Sloane's Database of Integer Sequences, Part 24
Part of the On-Line Encyclopedia of Integer Sequences
This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.
For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )
Seis.html: Welcome
index.html: Lookup
indexfr.html: Francais
demo1.html: Demos
Sindx.html: Index
WebCam.html: WebCam
Submit.html: Contribute new sequence or comment
eishelp1.html: Internal format
eishelp2.html: Beautified format
transforms.html: Transforms
Spuzzle.html: Puzzles
Shot.html: Hot
classic.html: Classics
ol.html: Superseeker
JIS/index.html: Journal of Integer Sequences
pages.html: More pages
Maintained by: N. J. A. Sloane (njas@research.att.com),
home page: www.research.att.com/~njas/
(start)
%I A016733
%S A016733 1,1,1,1,1,3,1,1,1,3,4,6,94,1,3,4,40,1,5,2,2,1,1,7,7,8,
%T A016733 39,4,3,4,1,9,1,1,2,1,7,1,6,2,18,1,1,1,12,1,1,3,1,1,4,16,
%U A016733 16,3,3,2,1,17,1,8,1,20,1,15,15,10,2,13,1,1,34,1,32,25
%N A016733 Continued fraction for ln(5).
%K A016733 nonn,cofr
%O A016733 1,6
%A A016733 njas
%I A060234
%S A060234 0,1,1,3,1,1,1,3,5,1,1,1,1,3,5,5,1,1,3,1,1,3,5,1,1,1,3,1,1,1,3,5,1,9,1,
%T A060234 1,1,3,5,5,1,1,1,1,1,7,7,3,1,1,5,1,1,5,5,5,1,1,1,1,3,13,3,1,1,9,1,7,1,
%U A060234 1,5,7,1,1,3,5,5,1,1,9,1,1,1,1,3,5,1,1,1,3,11,7,3,3,3,5,5,1,1,1,7,5,5
%N A060234 Residue of a prime modulo its distance from the successor prime: Mod[A000040(n),A001223(n)]=a(n).
%F A060234 Mod[Prime[w],Prime[w+1]-Prime[w]]
%e A060234 This residue is always odd: 3=1.2+1, 7=1.4+3, 23=3.6+5, etc...
%Y A060234 A000040, A001223.
%K A060234 nonn
%O A060234 0,4
%A A060234 Labos E. (labos@ana1.sote.hu), Mar 21 2001
%I A016466
%S A016466 3,1,1,1,3,6,2,19,2,2,7,1,2,18,7,1,4,1,1,1,4,1,6,2,2,1,
%T A016466 1,1,6,4,2,9,2,5,2,17,2,3,1,15,1,3,2,9,1,8,1,11,2,1,3,1,
%U A016466 1,125,1,1,7,1,36,1,1,5,1,18,1,7,1,103,49,20,1,2,1,2,6
%N A016466 Continued fraction for ln(38).
%K A016466 nonn,cofr
%O A016466 1,1
%A A016466 njas
%I A055210
%S A055210 1,1,1,3,1,1,1,3,7,1,1,3,1,1,1,11,1,7,1,3,1,1,1,3,21,1,7,3,1,1,1,11,1,
%T A055210 1,1,21,1,1,1,3,1,1,1,3,7,1,1,11,43,21,1,3,1,7,1,3,1,1,1,3,1,1,7,43,1,
%U A055210 1,1,3,1,1,1,21,1,1,21,3,1,1,1,11,61,1,1,3,1,1,1,3,1,7,1,3,1,1,1,11,1
%N A055210 Sum of totients of square divisors of n.
%F A055210 a(n) = Sum[Phi[d];d is square and divids n]
%F A055210 Multiplicative with a(p^e) = (p^(e+1)+1)/(p+1) for even e and a(p^e) = (p^e+1)/(p+1) for odd e. - Vladeta Jovovic (vladeta@Eunet.yu), Dec 01 2001
%e A055210 n = 400: its square divisors are {1, 4, 16, 25, 100, 400}, their totients are {1, 2, 8, 20, 40, 160} and the totient-sum over these divisors is, so a(400) = 231.This value arises at special square free multiples of 400 (400 times 2, 3, 5, 6, 7, 10, 11, 13, 15, 17, 19, 21, 22, 23 etc.)
%e A055210 a(400) = a(2^4*5^2) = (2^5+1)/3*(5^3+1)/6 = 231.
%K A055210 nonn,mult
%O A055210 1,4
%A A055210 Labos E. (labos@ana1.sote.hu), Jun 19 2000
%I A059619
%S A059619 1,1,1,1,0,1,3,1,1,1,4,2,0,1,1,6,2,1,1,1,1,10,4,2,1,1,1,1,15,6,3,1,2,1,
%T A059619 1,1,21,9,4,2,1,2,1,1,1,30,12,6,3,2,2,2,1,1,1,43,18,8,5,3,2,2,2,1,1,1,
%U A059619 59,25,12,6,3,3,3,2,2,1,1,1,82,34,17,9,5,4,3,3,2,2,1,1,1,111,48,22,12
%N A059619 As upper right triangle, number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing) where initial part is k.
%F A059619 T(n,k)=S(n,k)+sum_j[T(n-k,j)] for j>k, where S(n,k)=A059607(n,k)=sum_j[S(n-k,j)] for k>j [note reversal] with S(0,0)=1.
%e A059619 Rows start: {1,1,1,3,4,6,...}, {1,0,1,2,...}, {1,1,0,...} etc. T(16,6)=8 since 16 can be written as 6+10, 6+9+1, 6+8+2, 6+7+3, 6+7+2+1, 6+5+4+1, 6+5+3+2, or 6+4+3+2+1 (but for example neither 6+6+4 nor 6+8+1+1 which are only weakly unimodal).
%Y A059619 Top row is A059618 and is sum of other rows (for n>0). Cf. A000009, A000041, A001523, A059607.
%K A059619 nonn,tabl
%O A059619 0,7
%A A059619 Henry Bottomley (se16@btinternet.com), Jan 31 2001
%I A028234
%S A028234 1,1,1,1,1,3,1,1,1,5,1,3,1,7,5,1,1,9,1,5,7,11,1,3,1,13,1,7,1,15,1,1,
%T A028234 11,17,7,9,1,19,13,5,1,21,1,11,5,23,1,3,1,25,17,13,1,27,11,7,19,29,
%U A028234 1,15,1,31,7,1,13,33,1,17,23,35,1,9,1,37,25,19,11,39,1,5,1,41,1,21
%N A028234 If n = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then a(n) = n/p_1^e_1.
%Y A028234 Equals n/A028233(n).
%K A028234 nonn,nice,easy
%O A028234 1,6
%A A028234 Marc Le Brun (mlb@well.com)
%I A052125
%S A052125 1,1,1,1,1,3,1,1,1,5,1,4,1,7,5,1,1,9,1,5,7,11,1,8,1,13,1,7,1,15,1,1,11,
%T A052125 17,7,9,1,19,13,8,1,21,1,11,9,23,1,16,1,25,17,13,1,27,11,8,19,29,1,20,
%U A052125 1,31,9,1,13,33,1,17,23,35,1,9,1,37,25,19,11,39,1,16,1,41,1,28,17,43
%N A052125 n/A053597(n).
%e A052125 Since A053597(6) = 2, the value of a(6) here is 6/2 = 3.
%Y A052125 Cf. A053597.
%K A052125 nonn
%O A052125 1,6
%A A052125 James A. Sellers (sellersj@math.psu.edu), Jan 21 2000
%I A046643
%S A046643 1,1,1,3,1,1,1,5,3,1,1,3,1,1,1,35,1,3,1,3,1,1,1,5,3,1,5,3,1,1,1,
%T A046643 63,1,1,1,9,1,1,1,5,1,1,1,3,3,1,1,35,3,3,1,3,1,5,1,5,1,1,1,3,1,
%U A046643 1,3,231,1,1,1,3,1,1,1,15,1,1,3,3,1,1,1,35,35,1,1,3,1,1,1,5,1,3
%N A046643 From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives numerator of b_n.
%F A046643 Sum_{b|d} b(d)b(n/d) = 1. Also b_{2^j} = A001790[ j ]/2^A005187[ j ].
%e A046643 b_1, b_2, ... = 1, 1/2, 1/2, 3/8, 1/2, 1/4, 1/2, 5/16, 3/8, 1/4, 1/2, 3/16, ...
%p A046643 b:=proc(n) option remember; local c,i,t1; if n = 1 then 1 else c:=1; t1:=divisors(n);
%p A046643 for i from 2 to nops(t1)-1 do c:=c-b(t1[ i ])*b(n/t1[ i ]); od; c/2; fi; end;
%K A046643 nonn,easy,frac,nice
%O A046643 1,4
%A A046643 njas
%I A016566
%S A016566 3,1,1,1,6,3,1,1,1,5,3,1,1,96,1,1,2,2,3,2,3,2,32,1,1,1,
%T A016566 10,7,2,8,2,5,11,1,1,2,3,1,3,1,6,5,11,2,6,3,3,1,1,3,1,2,
%U A016566 1,1,2,2,8,3,1,2,1,1,1,5,6,1,4,3,187,1,3,1,6,1,1,1,6,3
%N A016566 Continued fraction for ln(77/2).
%K A016566 nonn,cofr
%O A016566 1,1
%A A016566 njas
%I A058057
%S A058057 1,1,0,1,1,0,1,3,1,1,1,6,6,8,3,1,10,20,38,35,16,1,15,50,134,213,
%T A058057 211,96,1,21,105,385,915,1479,1459,675,1,28,196,952,3130,7324,11692,
%U A058057 11584,5413,1,36,336,2100,9090,28764,65784,104364,103605,48800
%N A058057 Triangle giving coefficients of menage hit polynomials.
%D A058057 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
%e A058057 1; 1,0; 1,1,0; 1,3,1,1; 1,6,6,8,3; ...
%p A058057 V:=proc(n) local k; add( binomial(2*n-k,k)*(n-k)!*(x-1)^k, k=0..n); end; W:=proc(r,s) coeff( V(r),x,s ); end; a:=(n,k)->W(n,n-k);
%Y A058057 Diagonals give A000271, A000426, A000222, A000386, A000450, A058085, A058086.
%K A058057 nonn,easy,nice,tabl
%O A058057 0,8
%A A058057 njas, Dec 02 2000
%I A005765 M2210
%S A005765 1,1,3,1,1,1,7,7,1,1,11,1,1,1,15,1,16,1,19,1,1,1,23,22,1,25,27,1,1,1,
%T A005765 31,1,1,1,35,1,1,1,39,1,1,1,43
%N A005765 Size of Doehlert-Klee design with n blocks.
%D A005765 Doehlert, D. H.; Klee, V. L.; Experimental designs through level reduction of the $d$-dimensional cuboctahedron. Discrete Math. 2 (1972), no. 4, 309-334.
%D A005765 R. G. Stanton and S. A. Vanstone, Further results on a problem of Doehlert and Klee, Utilitas Math., 12 (1977), 263-271.
%K A005765 nonn
%O A005765 2,3
%A A005765 njas
%I A064085
%S A064085 1,1,1,1,1,3,1,1,1,11,1,39,1,43,151,1,1,171,1,2255,2359,683,1,9399,1,
%T A064085 2731,1,140911,1,1649373,1
%N A064085 Quotient of A000225 and A064084.
%C A064085 A064085(n) is equal to 1 if and only if n is a prime power; the sequence of non-trivial values of A064085 is A064086.
%F A064085 A064085(n) := A000225(n) / A064084(n)
%Y A064085 Cf. A000225, A064084, A064086.
%K A064085 easy,nonn
%O A064085 1,6
%A A064085 Jens Voss (jens.voss@poet.de), Sep 04 2001
%I A010278
%S A010278 3,1,1,1,14,2,2,1,34,2,1,2,1,1,6,4,1,1,2,1,80,4,1,3,3,1,
%T A010278 3,5,1,4,1,39,1,1,2,2,1,7,2,29,4,5,1,1,4,13,13,63,1,31,
%U A010278 3,1,2,4,1,17,19,4,2,16,5,2,1,3,1,2,4,4,5,4,5,4,1,3,3,1
%N A010278 Continued fraction for cube root of 49.
%K A010278 nonn,cofr
%O A010278 0,1
%A A010278 njas
%I A016467
%S A016467 3,1,1,1,35,8,2,130,2,2,3,1,1,1,5,15,11,1,11,3,9,1,7,1,
%T A016467 3,1,4,2,13,1,1,5,3,9,5,3,1,118,14,3,20,1,6,8,10,1,1,1,
%U A016467 1,1,1,1,1,1,1,2,1,1,1,3,1,13,46,3,1,4,10,1,2,1,7,9,13
%N A016467 Continued fraction for ln(39).
%K A016467 nonn,cofr
%O A016467 1,1
%A A016467 njas
%I A030347
%S A030347 1,3,1,1,2,1,1,5,2,2,1,1,1,1,2,1,1,2,1,1,1,3,1,1,1,2,1,1,3,2,
%T A030347 2,7,3,3,2,2,2,1,1,1,3,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,2,3,2,1,
%U A030347 1,1,1,1,1,2,1,1,2,1,1,1,3,1,1,1,2,1,1,1,2,1,1,1,1,2,4,1,2,1
%N A030347 Length of n-th run of digit 1 in A030341.
%K A030347 nonn
%O A030347 1,2
%A A030347 Clark Kimberling, ck6@cedar.evansville.edu
%I A010275
%S A010275 3,1,1,2,1,1,23,1,2,1,9,7,1,1,9,1,4,3,1,1,1,1,1,2,16,1,
%T A010275 1,1,3,8,2,4,1,2,2,9,4,11,1,11,1,9,1,1,1,7,5,14,1,1,1,8,
%U A010275 34,4,5,1,3,1,8,1,28,16,1,1,2,1,1,6,1,14,1,1,1,27,6,1,14
%N A010275 Continued fraction for cube root of 46.
%K A010275 nonn,cofr
%O A010275 0,1
%A A010275 njas
%I A050169
%S A050169 1,1,1,1,3,1,1,2,2,1,1,5,10,5,1,1,3,5,5,3,1,1,7,7,35,7,7,1,1,4,28,14,
%T A050169 14,28,4,1,1,9,12,42,126,42,12,9,1,1,5,15,30,42,42,30,15,5,1,1,11,55,
%U A050169 165,66,462,66,165,55,11,1,1,6,22,55,99,132,132,99,55,22,6,1,1,13,26
%N A050169 Triangular array T by rows: T(n,k)=GCD{C(n,k),C(n,k-1)), n >= 1, 1<=k<=n.
%C A050169 a(2n,n)=nth Catalan number; see A000108.
%F A050169 Also, T(n,k)=GCD{C(n,k),C(n+1,k).
%e A050169 Rows: {1}; {1,1}; {1,3,1}; ...
%K A050169 nonn,tabl
%O A050169 0,5
%A A050169 Clark Kimberling, ck6@cedar.evansville.edu
%I A064048
%S A064048 1,1,3,1,1,2,2,4,5,9,9,2,2,2,4,5,5,1,1,1,1,1,1,5,5,5,5,7,7,2,2,2,2,2,2,
%T A064048 4,4,4,4,1,1,1,1,1,3,3,3,4,4,4,4,4,4,4,4,6,6,6,6,4,4,4,4,4,4,4,4,4,4,6,
%U A064048 6,1,1,1,1,1,1,1,1,3,3,3,3,6,6,6,6,6,6,2,2,2,2,2,2,2,2,2,2,2
%N A064048 Number of most frequently occurring numbers in the 1-to-n multiplication table.
%e A064048 In the 1-to-6 multiplication table, the most frequently occurring numbers (each occurring 4 times) are 6 and 12. Therefore a(6)=2.
%Y A064048 Cf. A064047, A057142, A057143, A057144.
%K A064048 nonn
%O A064048 1,3
%A A064048 Matthew Somerville (matthew.somerville@trinity.oxford.ac.uk), Aug 24 2001
%I A016464
%S A016464 3,1,1,2,2,36,1,660,1,1,6,10,1,1,1,2,1,2,1,2,2,7,2,11,1,
%T A016464 2,1,1,1,2,2,1,76,3,1,1,6,6,11,2,4,1,2,18,1,1,1,1,1,15,
%U A016464 3,1,24,2,3,1,13,1,14,1,4,2,4,3,3,1,3,1,3,1,19,1,2,2,6
%N A016464 Continued fraction for ln(36).
%K A016464 nonn,cofr
%O A016464 1,1
%A A016464 njas
%I A030727
%S A030727 3,1,1,2,3,1,3,5,2,6,6,4,7,1,1,9,5,8,1,2,3,1,12,7,10,2,4,4,2,
%T A030727 1,1,15,10,11,5,5,5,4,2,2,1,1,18,13,12,7,9,6,5,3,3,3,1,2,1,21,
%U A030727 15,16,8,11,8,7,4,5,4,2,4,1,2,1,24,18,17,12,13,9,9,7,6,5,4,5
%N A030727 Row 1, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in rows 1 and 2, for j=1,2,...,m, where m=max number in row 1 from stages 1 to k-1; stage 1 is 3 in row 1.
%K A030727 nonn
%O A030727 1,1
%A A030727 Clark Kimberling, ck6@cedar.evansville.edu
%I A065836
%S A065836 3,1,1,2,3,3,2,3,2,3,3,3,2,0,2,1,1,3,3,1,0,2,0,2,3,0,1,0,2,2,0,2,0,3,2,
%T A065836 3,2,1,1,0,2,1,0,1,3,2,2,3,0,0,3,0,0,2,2,3,1,2,3,0,1,2,1,1,1,0,2,1,0,2,
%U A065836 0,1,3,2,1,1,0,2,2,3,0,3,1,2,1,0,3,3,1,2,2,3,3,3,1,2,1,2,0,1,0,1,2,3,0
%N A065836 Quaternary digits found in decimal expansion of pi.
%Y A065836 Cf. A065828 up to A065840, A000796, A011545, A011546.
%K A065836 nonn,base
%O A065836 0,1
%A A065836 Patrick De Geest (pdg@worldofnumbers.com), Nov 24 2001.
%I A035254
%S A035254 3,1,1,2,4,5,6,8,9,10,11,13,14,15,16,17,19,20,21,22,23,
%T A035254 24,26,27,28,29,30,31,32,34,35,36,37,38,39,40,41,43,44,
%U A035254 45,46,47,48,49,50,51,53,54,55,56,57,58,59,60,61,62,64
%V A035254 -3,-1,1,2,4,5,6,8,9,10,11,13,14,15,16,17,19,20,21,22,23,
%W A035254 24,26,27,28,29,30,31,32,34,35,36,37,38,39,40,41,43,44,
%X A035254 45,46,47,48,49,50,51,53,54,55,56,57,58,59,60,61,62,64
%N A035254 First differences of A035253. First differences are A035214.
%Y A035254 Different from A030124. Cf. A014132, A014133.
%K A035254 sign,done,easy
%O A035254 0,1
%A A035254 Robet Bronson (bob@bronsons.com)
%I A016564
%S A016564 3,1,1,2,14,2,13,1,2,2,1,14,1,106,2,1,3,5,1,51,1,1,2,4,
%T A016564 1,13,1,6,1,1,18,1,2,1,1,3,33,3,1,3,1,3,5,8,1,126,1,9,1,
%U A016564 1,16,4,1,2,1,4,7,1,1,17,1,1,3,1,2,6,1,3,1,2,1,1,2,1,3
%N A016564 Continued fraction for ln(73/2).
%K A016564 nonn,cofr
%O A016564 1,1
%A A016564 njas
%I A058965
%S A058965 0,3,1,1,3,1,1,1,1,3,1,3,12,1,8,8,1,7,6,1,5,2,1,1,4,1,3,2,36,1,
%T A058965 10,6,1,2
%N A058965 Continued fraction expansion of series-parallel constant.
%D A058965 J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
%D A058965 J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.
%F A058965 This number, c, is defined by Product_{n=1..inf} (1-c^n)^(-A000669[n]) = 2.
%e A058965 .2808326669842003553932...
%Y A058965 See A058964 for decimal expansion. Cf. A000084, A000669.
%K A058965 nonn,cofr
%O A058965 0,2
%A A058965 njas, emr Jan 14 2001
%I A049653
%S A049653 1,1,1,3,1,1,3,1,1,3,1,3,5,1,1,3,5,1,3,1,1,3,1,3,5,1,3,5,1,1,3,
%T A049653 5,1,3,1,1,3,5,1,3,1,3,5,1,3,5,7,1,3,1,1,3,1,1,3,1,3,5,7,9,11,13,
%U A049653 1,3,1,3,5,1,1,3,5,7,9,1,1,3,5,1,3,5,1,3,1,3,5,1,3,5,1,1,3,5,7
%N A049653 2*n-prevprime(2*n).
%Y A049653 Cf. A049613, A049711, A049716, A049847.
%K A049653 nonn
%O A049653 2,4
%A A049653 njas
%I A060266
%S A060266 1,1,1,3,1,1,3,1,1,3,1,5,3,1,1,5,3,1,3,1,1,3,1,5,3,1,5,3,1,1,5,3,1,3,1,
%T A060266 1,5,3,1,3,1,5,3,1,7,5,3,1,3,1,1,3,1,1,3,1,13,11,9,7,5,3,1,3,1,5,3,1,1,
%U A060266 9,7,5,3,1,1,5,3,1,5,3,1,3,1,5,3,1,5,3,1,1,9,7,5,3,1,1,3,1,1,11,9,7,5
%N A060266 Difference between 2n and the following prime.
%p A060266 with(numtheory): [seq(nextprime(2*i)-2*i,i=1..256)];
%Y A060266 Cf. A020482, A049653, A060266-A060268, A060264.
%K A060266 nonn
%O A060266 1,4
%A A060266 Labos E. (labos@ana1.sote.hu), Mar 23 2001
%I A046929
%S A046929 0,0,1,1,1,1,1,1,3,1,1,3,1,1,3,5,1,1,3,1,1,3,3,5,3,1,1,1,1,3,3,3,
%T A046929 1,1,1,1,5,3,3,5,1,1,1,1,1,1,11,3,1,1,3,1,1,5,5,5,1,1,3,1,1,9,3,
%U A046929 1,1,3,5,5,1,1,3,5,5,5,3,3,5,3,3,7,1,1,1,1,3,3,5,3,1,1,3,7,3,3
%N A046929 Width of moat of composite numbers surrounding n-th prime.
%e A046929 23 has a buffer of 3 composites around it on each side: 20,21,22,23,24,25,26.
%p A046929 with(numtheory); a:=i->min(ithprime(n)-ithprime(n-1)-1, ithprime(n+1)-ithprime(n)-1);
%Y A046929 Related sequences: A023186-A023188, A046929-A046931, A051650, A051652, A051697-A051702, A051728-A051730.
%K A046929 nonn,easy,nice
%O A046929 1,9
%A A046929 njas
%I A038500
%S A038500 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,27,
%T A038500 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,27,
%U A038500 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,81
%N A038500 Highest power of 3 dividing n.
%Y A038500 Cf. A007949.
%K A038500 nonn
%O A038500 1,3
%A A038500 njas
%I A046111
%S A046111 1,1,1,3,1,1,3,1,3,1,1,3,1,3,1,1,5,3,1,3,1,1,3,3,3,1,3,3,1,1,1,1,1,5,3,
%T A046111 3,3,1,3,1,3,1,1,9,1,3,3,1,3,3,1,1,1,3,3,1,9,1,1,3,3,1,1,3,3,1,5,3,1,3,
%U A046111 3,1,3,3,1,3,3,3,1,3,1,3,1,7,1,1,9,1,1,1,3,3,1,3,1,3,9,3,3,3,1,1
%N A046111 Number of lattice points on circumference of a circle of radius 1/3,2/3,4/3,5/3,... with center at (1/3,0).
%H A046111 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A046111 Cf. A046109.
%K A046111 nonn
%O A046111 0,4
%A A046111 Eric W. Weisstein (eric@weisstein.com)
%I A035691
%S A035691 0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,3,1,1,3,1,3,3,2,6,3,4,7,4,8,7,6,13,8,
%T A035691 10,15,10,17,17,14,24,19,22,30,23,33,34,31,46,39,44,56,47,63,65,61,82,
%U A035691 75,84,101,90,113,118,115,145,137,151,176,165,197,207,206,246,242,264
%N A035691 Partitions into parts 8k+3 and 8k+5 with at least one part of each type
%K A035691 nonn,part
%O A035691 1,16
%A A035691 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A010274
%S A010274 3,1,1,3,1,8,2,4,1,1,2,5,1,5,1,1,1,20,1,1,1,1,1,5,1,1,1,
%T A010274 48,4,2,1,1,2,2,3,2,1,2,1,1,1,1,1,1,187,3,1,3,6,6,1,2,10,
%U A010274 1,1,5,2,1,1,1,13,1,61,1,4,1,1,1,1,22,2,4,3,2,1,1,1,3,52
%N A010274 Continued fraction for cube root of 45.
%K A010274 nonn,cofr
%O A010274 0,1
%A A010274 njas
%I A054398
%S A054398 1,1,3,1,1,3,1,11,1,3,1,1,3,1,11,1,3,1,43,1,3,1,11,1,3,1,1,3,1,11,1,3,
%T A054398 1,43,1,3,1,11,1,3,1,171,1,3,1,11,1,3,1,43,1,3,1,11,1,3,1,1,3,1,11,1,3,
%U A054398 1,43,1,3,1,11,1,3,1,171,1,3,1,11,1,3,1,43,1,3,1,11,1,3,1,683,1,3,1,11
%N A054398 Define a sequence of 2^n X 2^n squares as follows: S_0 = [1], S_1 = [1,2; 3,4]; S_2 = [1,2,5,6; 3,4,7,8; 9,10,13,14; 11,12,15,16], etc.; sequence gives triangular array whose n-th row gives differences between successive columns of n-th square.
%e A054398 1; 1,3,1; 1,3,1,11,1,3,1; ...
%K A054398 nonn,tabf,easy,nice
%O A054398 1,3
%A A054398 Tidjani Negadi (t-negadi@usa.net), May 21 2000
%E A054398 More terms from Naohiro Nomoto (6284968128@geocities.co.jp), Sep 12 2001
%I A023142
%S A023142 1,1,3,1,1,3,2,1,9,1,6,3,3,2,3,1,2,9,2,1,6,6,2,3,1,3,15,2,2,3,3,1,18,2,2,
%T A023142 9,13,2,9,1,9,6,3,6,9,2,2,3,3,1,6,3,5,15,6,2,6,2,2,3,2,3,18,1,3,18,3,2,6,
%U A023142 2,3,9,10,13,3,2,17,9,7,1,21,9,3,6,2,3,6,6,3,9,16,2,9,2,2,3,2,3,54,1,26
%N A023142 Number of cycles of function f(x) = 10x mod n.
%K A023142 nonn
%O A023142 1,3
%A A023142 dww
%I A033989
%S A033989 0,3,1,1,3,2,7,9,1,1,6,9,4,7,9,1,2,1,2,1,6,7,4,3,6,1,2,9,5,1,1,0,9,3,1,
%T A033989 3,6,6,1,8,6,9,2,5,0,2,2,4,6,6,2,5,6,0,3,8,9,5,3,3,6,9,4,0,5,4,4,9,8,0,
%U A033989 5,0,4,5,5,3,3,1,6,8,5,8,6,5,1,4,7,4,9,1,8,5,1,9,9,8,6,6,9,1,1,6,4,8,1
%N A033989 Write 0,1,2,... in clockwise spiral, writing each digit in separate square; sequence gives numbers on negative x axis.
%e A033989 131416...
%e A033989 245652...
%e A033989 130717...
%e A033989 121862...
%e A033989 101918...
%Y A033989 Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%K A033989 nonn,easy,nice
%O A033989 0,2
%A A033989 njas
%E A033989 More terms from Andrew J. Gacek (andrew@dgi.net)
%I A046540
%S A046540 1,1,1,1,3,1,1,3,3,1,1,3,3,3,1,1,3,1,1,3,1,1,3,3,1,3,3,1,1,3,3,3,3,3,3,
%T A046540 1,1,3,1,3,3,3,1,3,1,1,3,3,3,3,3,3,3,3,1,1,3,3,3,3,3,3,3,3,3,1
%N A046540 Denominators of the 1/3-Pascal triangle (by row).
%e A046540 1/1; 1/1 1/1; 1/1 1/3 1/1; 1/1 4/3 4/3 1/1; 1/1 7/3 8/3 7/3 1/1; 1/1 10/3 5/1 5/1 10/3 1/1; 1/1 13/3 25/3 10/1 25/3 13/3 1/1; 1/1 16/3 38/3 55/3 55/3 38/3 16/3 1/1; ...
%Y A046540 Cf. A046534.
%K A046540 tabl,nonn
%O A046540 1,5
%A A046540 Mohammad K. Azarian, ma3@cedar.evansville.edu
%I A025834
%S A025834 1,0,0,1,1,0,1,1,1,1,1,1,3,1,1,3,3,1,3,3,3,3,3,3,6,3,3,
%T A025834 6,6,3,6,6,6,6,6,6,10,6,6,10,10,6,10,10,10,10,10,10,15,
%U A025834 10,10,15,15,10,15,15,15,15,15,15,21,15,15,21,21,15,21
%N A025834 Expansion of 1/((1-x^3)(1-x^4)(1-x^12)).
%K A025834 nonn
%O A025834 0,13
%A A025834 njas
%I A035649
%S A035649 0,0,0,0,0,0,1,0,0,1,1,0,3,1,1,3,3,1,7,3,3,8,8,3,14,9,8,16,17,9,27,19,
%T A035649 18,32,34,20,49,40,37,58,63,43,87,74,70,104,113,82,149,135,128,177,
%U A035649 195,152,249,232,224,298,327,266,407,392,380,485,535,455,654,639,628
%N A035649 Partitions into parts 6k+3 and 6k+4 with at least one part of each type
%K A035649 nonn,part
%O A035649 1,13
%A A035649 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A035666
%S A035666 0,0,0,0,0,0,1,0,0,1,1,0,1,3,1,1,3,3,2,3,6,4,4,7,8,6,8,13,10,10,16,17,
%T A035666 14,19,25,22,23,32,34,31,38,48,45,47,60,65,62,73,86,86,90,109,117,117,
%U A035666 133,153,155,165,191,205,209,235,261,272,288,326,349,362,398,440,459
%N A035666 Partitions into parts 7k+3 and 7k+4 with at least one part of each type
%K A035666 nonn,part
%O A035666 1,14
%A A035666 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A060592
%S A060592 0,1,1,1,0,1,3,1,1,3,3,3,0,3,3,2,3,2,2,3,2,3,2,2,0,2,2,3,2,2,1,1,1,1,2,
%T A060592 2,3,1,3,1,0,1,3,1,3,7,2,2,1,1,1,1,2,2,7,6,6,3,2,2,0,2,2,3,6,6,7,5,7,2,
%U A060592 3,2,2,3,2,7,5,7,6,6,6,6,3,3,0,3,3,6,6,6,6,7,5,7,5,5,3,1,1,3,5,5,7,5,7
%N A060592 Square table by anti-diagonals of minimum number of moves between two positions in the Tower of Hanoi (with three pegs: 0,1,2), where with position n written in base 3, xyz means smallest disk is on peg z, second smallest is on peg y, third smallest on peg x, etc. and leading zeros indicate largest disks are all on peg 0.
%H A060592 Graph of positions and possible routes on a Sierpinski triangle
%e A060592 T(4,9)=5 since 4 and 9 written in base 3 are 11 and 100, i.e. the starting position has the first and second disks on peg 1 and the others on peg 0, while the end position has the third disk on peg 1 and the others on peg 0; the five optimal moves between these positions are: move the third disk to peg 2, then the first to peg 2, the second to peg 0, the first to peg 0, and finally the third to peg 1.
%Y A060592 Cf. A001511, A007798, A055661, A055662.
%K A060592 nonn,tabl
%O A060592 0,7
%A A060592 Henry Bottomley (se16@btinternet.com), Apr 06 2001
%I A059959
%S A059959 1,1,1,1,3,1,1,3,3,5,3,1,3,3,1,1,3,1,3,9,3,9,3,35,5,29,3,3,3,1,5,9,25,
%T A059959 31,5,9,7,7,15,21,11,29,7,55,15,5,21,69,27,21,21,5,33,3,5,9,27,55,33,1,
%U A059959 57,25,13,49,5,3,23,19,25,11,15,29,35,33,15,11,7,23,13,17,9,55,3,19,27
%N A059959 Distance of 2^n from its nearest prime neighbor, either below or above.
%e A059959 n=19, 2^19=524288, prevprime(524288)=524287, nextprime(524288)=524309,so min{21,1}=1=a(19)
%p A059959 with(numtheory): [seq(min(nextprime(2^i)-2^i,2^i-prevprime(2^i)),i=2..100)];
%Y A059959 A013597, A013603, A014210, A014234, A058249.
%K A059959 nonn
%O A059959 2,5
%A A059959 Labos E. (labos@ana1.sote.hu), Mar 02 2001
%I A051120
%S A051120 1,1,1,3,1,1,3,4,1,1,4,2,3,6,1,1,6,2,4,3,3,1,2,8,1,1,8,2,5,3,4,5,3,3,2,
%T A051120 11,1,1,11,2,8,3,6,1,5,4,4,8,3,5,2,13,1,1,13,2,11,4,8,4,6,3,5,6,4,10,3,
%U A051120 7,2,16,1,1,16,2,13,3,11,1,10,5,8,1,7,6,6,4,5,9,4,12,3,9,2,18,1
%N A051120 Start with 1; at n-th step, write down what is in the sequence so far.
%C A051120 After 1 1 1 3 1, we see "1 3 and 4 1's", so next terms are 1 3 4 1. Then "1 4, 2 3's, 6 1's"; etc.
%K A051120 nice,nonn,base,easy
%O A051120 0,4
%A A051120 Jamie (sunshinebaby@hotmail.com)
%E A051120 More terms from Michael Lugo (mlugo@thelabelguy.com), Dec 22 1999
%I A035690
%S A035690 0,0,0,0,0,0,1,0,0,1,1,0,1,1,3,1,1,3,4,1,3,4,7,3,4,8,10,4,8,11,15,8,
%T A035690 11,18,21,11,19,24,30,19,25,37,42,25,40,50,56,41,53,70,79,54,77,95,
%U A035690 103,80,103,129,141,106,144,172,183,151,189,228,246,197,257,301,314
%N A035690 Partitions into parts 8k+3 and 8k+4 with at least one part of each type
%K A035690 nonn,part
%O A035690 1,15
%A A035690 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A027960
%S A027960 1,1,3,1,1,3,4,4,1,1,3,4,7,8,5,1,1,3,4,7,11,15,13,6,1,1,3,4,7,
%T A027960 11,18,26,28,19,7,1,1,3,4,7,11,18,29,44,54,47,26,8,1,1,3,4,7,
%U A027960 11,18,29,47,73,98,101,73,34,9,1,1,3,4,7,11,18,29,47,76,120,171
%N A027960 Triangular array T by rows: T(n,0)=T(n,2n)=1 for n >= 0; T(n,1)=3 for n >= 1; and for n >= 2, T(n,k)=T(n-1,k-2)+T(n-1,k-1) for k=2,3,...,2n-1.
%F A027960 T(n,k) = number of strings s(0),...,s(n) such that s(n)=n-k. s(0) in {0,1,2}, s(1)=1 if s(0) in {1,2}, s(1) in {0,1,2} if s(0)=0, and for 1<=i<=n, s(i)=s(i-1)+d, with d in {0,2} if s(i)=2i, in {0,1,2} if s(i)=2i-1, in {0,1} if 0<=s(i)<=2i-2.
%K A027960 nonn
%O A027960 1,3
%A A027960 Clark Kimberling, ck6@cedar.evansville.edu
%I A016599
%S A016599 3,1,1,3,5,1,5,3,0,9,2,1,0,3,7,4,4,4,7,9,7,4,0,1,7,6,8,5,6,1,3,0,6,
%T A016599 2,4,8,0,7,4,5,0,8,2,3,3,5,5,5,3,7,6,1,3,7,1,2,6,1,3,5,6,5,4,9,2,5,
%U A016599 5,7,7,3,9,5,2,1,7,5,1,8,0,9,8,2,7,7,1,5,0,2,0,6,0,7,2,4,2,2,5,1,0
%N A016599 Decimal expansion of ln(45/2).
%K A016599 nonn,cons
%O A016599 1,1
%A A016599 njas
%I A058735
%S A058735 1,1,1,1,1,3,1,1,3,7,1,1,3,8,20,1,1,3,8,22,55,1,1,3,8,23,63,162,1,1,
%T A058735 3,8,23,65,188,477
%N A058735 Triangle of coefficients in expansion of enumerators for series-reduced rooted trees by lines at the root.
%D A058735 J. Riordan, The blossoming of Schroeder's fourth problem, Acta Math., 137 (1976), 1-16.
%H A058735 Index entries for sequences related to rooted trees
%e A058735 1; 1,1; 1,1,3; 1,1,3,7; ...
%Y A058735 Main diagonal is A058737.
%K A058735 nonn,tabl
%O A058735 0,6
%A A058735 njas, Jan 01 2001
%I A011085
%S A011085 3,1,1,3,7,3,7,2,5,8,4,7,7,7,6,9,8,3,3,7,0,6,4,7,2,7,9,2,0,6,7,1,2,
%T A011085 7,3,8,1,3,0,1,8,5,8,1,9,8,4,8,5,6,2,9,0,5,9,6,8,2,0,3,9,9,7,2,3,7,
%U A011085 8,3,7,2,5,8,7,3,2,8,6,0,2,2,8,9,5,1,5,8,7,1,0,8,9,6,0,0,3,5,9,7,7
%N A011085 Decimal expansion of 4th root of 94.
%K A011085 nonn,cons
%O A011085 1,1
%A A011085 njas
%I A024737
%S A024737 3,1,1,3,9,27,84,261,810,2514,8046,25704,81975,261261,854037,2784540,
%T A024737 9057393,29439855,97650930,323099676,1066565907,3517004046,11791407663,
%U A024737 39431809098,131550859650,438453827928,1481254263945,4993010577093
%N A024737 a(n) = Sum (a(2i-1)*a(n-2i+1), i = 1,2,...,k), where k = [ (n+1)/4 ].
%K A024737 nonn
%O A024737 1,1
%A A024737 Clark Kimberling (ck6@cedar.evansville.edu)
%I A024958
%S A024958 3,1,1,3,9,28,87,270,838,2682,8568,27325,87087,284940,929773,3026424,
%T A024958 9843471,32691629,108290562,357839667,1181136576,3967571310,13290977466,
%U A024958 44410377199,148240940865,501864577470,1694984858142,5711630686332
%N A024958 a(n) = SUM(a(2i-1)*a(n-2i+1), i = 1,2,...,[ (n+2)/4 ]).
%K A024958 nonn
%O A024958 4,1
%A A024958 Clark Kimberling (ck6@cedar.evansville.edu)
%I A014516
%S A014516 3,1,1,3,9,40,214,1355,9940,82834,773118,7988888,90540730,
%T A014516 1116669003,14888920041,213407853922,3272253760146,53446811415729,
%U A014516 926411397872639,16984208960998389,328361373245968867
%N A014516 Nearest integer to GAMMA(n+1/3).
%D A014516 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 255.
%p A014516 [ seq(round(evalf(GAMMA(n+1/3),100)),n=0..24) ];
%K A014516 nonn
%O A014516 0,1
%A A014516 njas
%I A016563
%S A016563 3,1,1,3,10,2,11,3,2,4,1,4,4,2,32,1,3,1,50,3,4,1,1,3,1,
%T A016563 1,1,3,1,3,2,3,3,17,1,4,5,3,5,1,6,3,1,5,1,10,2,1,4,3,1,
%U A016563 3,3,1,3,2,1,2,9,26,1,1,20,1,1,5,1,1,4,1,2,5,1,3,5,1,1
%N A016563 Continued fraction for ln(71/2).
%K A016563 nonn,cofr
%O A016563 1,1
%A A016563 njas
%I A025254
%S A025254 3,1,1,3,10,34,118,417,1497,5448,20063,74649,280252,1060439,4040413,
%T A025254 15488981,59701236,231236830,899559100,3513314664,13770811198,54152480421,
%U A025254 213585706927,844723104691,3349274471386,13310603555085,53012829376985
%N A025254 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.
%K A025254 nonn
%O A025254 1,1
%A A025254 Clark Kimberling (ck6@cedar.evansville.edu)
%I A000503
%S A000503 0,1,3,1,1,4,1,0,7,1,0,226,1,0,7,1,0,3,2,0,2,2,0,1,3,1,1,
%T A000503 4,1,0,7,1,0,76,1,0,7,1,0,3,2,0,2,2,0,1,3,1,1,4,1,0,7,1,0,
%U A000503 46,1,0,8,1,0,3,2,0,2,2,0,1,3,1,1,4,1,0,6,1,0,33,1,0,9,1
%V A000503 0,1,-3,-1,1,-4,-1,0,-7,-1,0,-226,-1,0,7,-1,0,3,-2,0,2,-2,0,1,-3,-1,1,
%W A000503 -4,-1,0,-7,-1,0,-76,-1,0,7,-1,0,3,-2,0,2,-2,0,1,-3,-1,1,-4,-1,0,-7,
%X A000503 -1,0,-46,-1,0,8,-1,0,3,-2,0,2,-2,0,1,-3,-1,1,-4,-1,0,-6,-1,0,-33,-1
%N A000503 [ tan(n) ].
%p A000503 f:=n->floor(evalf(tan(n)));
%Y A000503 Cf. A005657.
%K A000503 sign,done,easy,nice
%O A000503 0,3
%A A000503 njas
%I A024564
%S A024564 1,3,1,1,4,1,1,6,1,2,9,1,2,24,1,3,1,1,3,1,1,4,1,2,6,1,2,12,1,2,54,1,3,1,1,4,
%T A024564 1,1,5,1,2,8,1,2,1,7,1,2,1,1,3,1,1,4,1,1,6,1,2,10,1,2,27,1,3,1,1,3,1,1,4,1,
%U A024564 2,7,1,2,12,1,2,69,1,3,1,1,4,1,1,5,1,2,8,1,2,18,1,2,1,1,3,1,1
%N A024564 a(n) = [ 1/{n*sqrt(7)} ], where {x} := x - [ x ].
%K A024564 nonn
%O A024564 1,2
%A A024564 Clark Kimberling (ck6@cedar.evansville.edu)
%I A030184
%S A030184 0,1,1,1,1,1,1,0,3,1,1,4,1,2,0,1,1,2,1,4,1,0,4,0,3,1,2,
%T A030184 1,0,2,1,0,5,4,2,0,1,10,4,2,3,10,0,4,4,1,0,8,1,7,1,2,2,
%U A030184 10,1,4,0,4,2,4,1,2,0,0,7,2,4,12,2,0,0,8,3,10,10,1,4,0
%V A030184 0,1,-1,-1,-1,1,1,0,3,1,-1,-4,1,-2,0,-1,-1,2,-1,4,-1,0,4,0,-3,1,2,
%W A030184 -1,0,-2,1,0,-5,4,-2,0,-1,-10,-4,2,3,10,0,4,4,1,0,8,1,-7,-1,-2,2,
%X A030184 -10,1,-4,0,-4,2,-4,1,-2,0,0,7,-2,-4,12,-2,0,0,-8,3,10,10,-1,-4,0
%N A030184 Expansion of eta(q)*eta(q^3)*eta(q^5)*eta(q^15).
%D A030184 N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 70.
%K A030184 sign,done
%O A030184 1,9
%A A030184 njas
%I A023579
%S A023579 1,1,3,1,1,4,2,1,1,5,1,3,2,1,1,3,1,6,1,1,2,1,1,2,2,3,1,1,4,2,1,1,2,
%T A023579 1,3,1,5,1,1,4,1,3,1,2,3,1,1,1,1,3,2,1,2,1,2,1,4,1,3,2,1,3,1,1,2,6,
%U A023579 1,2,1,5,2,1,1,3,1,1,3,4,2,2,1,3,1,2,1,1,2,2,4,1,1,1,1,1,1,1,9,2,1
%N A023579 Exponent of 2 in prime factorization of p(n)+3.
%K A023579 nonn
%O A023579 1,3
%A A023579 Clark Kimberling (ck6@cedar.evansville.edu)
%I A023577
%S A023577 1,1,3,1,1,4,2,1,1,5,1,3,2,1,2,3,1,6,1,1,2,1,1,2,2,3,1,1,4,2,1,1,2,
%T A023577 1,3,1,5,1,1,4,1,3,1,2,3,1,1,1,1,3,2,2,2,1,2,1,4,1,3,2,1,3,1,1,2,6,
%U A023577 1,2,2,5,2,1,1,3,1,1,3,4,2,2,1,3,1,2,1,1,2,2,4,1,1,1,2,1,1,1,9,2,1
%N A023577 Greatest exponent in prime-power factorization of p(n)+3.
%K A023577 nonn
%O A023577 1,3
%A A023577 Clark Kimberling (ck6@cedar.evansville.edu)
%I A049999
%S A049999 1,1,1,1,1,1,3,1,1,4,3,1,5,4,1,1,6,5,1,3,7,6,1,1,4,8,7,3,1,5,9,8,4,1,1,
%T A049999 6,10,9,5,1,3,7,11,10,6,1,1,4,8
%N A049999 a(n)=index k such that F(k)=d(n), where d=A049997 (difference sequence of ordered products of Fibonacci numbers).
%K A049999 nonn
%O A049999 1,7
%A A049999 Clark Kimberling, ck6@cedar.evansville.edu
%I A036040
%S A036040 1,1,1,1,3,1,1,4,3,6,1,1,5,10,10,15,10,1,1,6,15,10,15,60,15,20,45,15,1,1,
%T A036040 7,21,35,21,105,70,105,35,210,105,35,105,21,1,1,8,28,56,35,28,168,280,
%U A036040 210,280,56,420,280,840,105,70,560,420,56,210,28,1,1,9,36,84,126,36,252
%N A036040 Triangle of multinomial coefficients.
%D A036040 Abramowitz and Stegun, Handbook, p. 831, column labeled "M_3".
%e A036040 1; 1,1; 1,3,1; 1,4,3,6,1; ...
%Y A036040 Cf. A036036-A036039.
%K A036040 nonn,easy,nice,tabf
%O A036040 1,5
%A A036040 njas
%E A036040 More terms from dww.
%I A049687
%S A049687 0,1,1,1,3,1,1,4,4,1,1,5,5,5,1,1,6,7,7,6,1,1,7,8,9,8,7,1,1,8,10,11,11,
%T A049687 10,8,1,1,9,11,14,13,14,11,9,1,1,10,13,15,17,17,15,13,10,1,1,11,14,18,
%U A049687 18,21,18,18,14,11,1,1,12,16,20,22,23,23,22,20,16,12,1,1,13,17,22,24
%N A049687 Array T by diagonals: T(i,j)=number of lines passing through (0,0) and at least one other lattice point (h,k) satisfying 0<=h<=i, 0<=k<=j.
%K A049687 nonn,tabl,nice
%O A049687 0,5
%A A049687 Clark Kimberling, ck6@cedar.evansville.edu
%E A049687 More terms from Michael Somos (somos@grail.cba.csuohio.edu)
%E A049687 Cf. A049639.
%I A028262
%S A028262 1,1,1,1,3,1,1,4,4,1,1,5,8,5,1,1,6,13,13,6,1,1,7,19,26,19,7,1,1,8,26,
%T A028262 45,45,26,8,1,1,9,34,71,90,71,34,9,1,1,10,43,105,161,161,105,43,10,1,1,
%U A028262 11,53,148,266,322,266,148,53,11,1,1,12,64,201,414,588,588,414,201,64
%N A028262 Elements in 3-Pascal triangle (by row).
%F A028262 After the 3rd row, use Pascal's rule.
%e A028262 1; 1 1; 1 3 1; 1 4 4 1; 1 5 8 5 1; ...
%K A028262 nonn,nice,tabl
%O A028262 0,5
%A A028262 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A028262 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A050177
%S A050177 1,1,1,1,3,1,1,4,4,1,1,5,9,5,1,1,6,14,14,6,1,1,7,20,28,20,7,1,1,8,27,
%T A050177 48,48,27,8,1,1,9,35,75,90,75,35,9,1,1,10,44,110,165,165,110,44,10,1,1,
%U A050177 11,54,154,275,297,275,154,54,11,1,1,12,65,208,429,572,572
%N A050177 T(n,k)=M0(n,k,f(n,k)), where M0 is given by A050176 and f(n,k) is the next-to-least t for which M0(n,k,t) is not 0.
%C A050177 f(n,k)=0 if 1<=k<=[ (n-1)/2 ], else f(n,k)=2k-n+1.
%e A050177 Rows: {1}; {1,1}; {1,3,1}; {1,4,4,1}; ... (all palindromes)
%K A050177 nonn,tabl
%O A050177 3,5
%A A050177 Clark Kimberling, ck6@cedar.evansville.edu
%I A013580
%S A013580 1,1,1,1,3,1,1,4,4,1,1,5,9,5,1,1,6,14,14,6,1,1,7,20,29,20,7,1,1,8,27,
%T A013580 49,49,27,8,1,1,9,35,76,99,76,35,9,1,1,10,44,111,175,175,111,44,10,1,1,
%U A013580 11,54,155,286,351,286,155,54,11,1,1,12,65,209,441,637,637,441,209,65
%N A013580 Triangle formed in same way as Pascal's triangle (A007318) except 1 is added to central element in even-numbered rows.
%e A013580 1; 1,1; 1,3,1; 1,4,4,1; 1,5,9,5,1; ...
%K A013580 tabl,nonn,easy
%O A013580 0,5
%A A013580 Martin Hecko (bigusm@interramp.com)
%E A013580 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A026670
%S A026670 1,1,1,1,3,1,1,4,4,1,1,5,11,5,1,1,6,16,16,6,1,1,7,22,43,22,7,
%T A026670 1,1,8,29,65,65,29,8,1,1,9,37,94,173,94,37,9,1,1,10,46,131,267,
%U A026670 267,131,46,10,1,1,11,56,177,398,707,398,177,56,11,1,1,12,67
%N A026670 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 1, T(n,1)=T(n,n-1)=n-1; for n >= 2, T(n,k)=T(n-1,k-1)+T(n-1,k-2)+T(n-1,k) if n is even and k=n/2, else T(n,k)=T(n-1,k-1)+T(n-1,k).
%F A026670 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,j)-to-(i+1,j+1) for i=j.
%K A026670 nonn,tabl
%O A026670 1,5
%A A026670 Clark Kimberling, ck6@cedar.evansville.edu
%E A026670 Formula corrected by David Perkinson (davidp@reed.edu), Sep 19 2001
%I A026626
%S A026626 1,1,1,1,3,1,1,4,4,1,1,6,8,6,1,1,7,14,14,7,1,1,9,21,28,21,9,1,
%T A026626 1,10,30,49,49,30,10,1,1,12,40,79,98,79,40,12,1,1,13,52,119,177,
%U A026626 177,119,52,13,1,1,15,65,171,296,354,296,171,65,15,1,1,16,80
%N A026626 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; T(n,1)=T(n,n-1)=[ 3n/2 ] for n >= 1; T(n,k)=T(n-1,k-1)+T(n-1,k) for 2<=k<=n-2, n >= 4.
%H A026626 Index entries for sequences related to rooted trees
%F A026626 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,j)-to-(i+1,j+1) for i=0, j >= 0 and even, and for j=0, i >= 0 and even.
%K A026626 nonn,tabl
%O A026626 1,5
%A A026626 Clark Kimberling, ck6@cedar.evansville.edu
%I A026648
%S A026648 1,1,1,1,3,1,1,4,4,1,1,6,8,6,1,1,7,14,14,7,1,1,9,21,36,21,9,1,
%T A026648 1,10,30,57,57,30,10,1,1,12,40,108,114,108,40,12,1,1,13,52,148,
%U A026648 222,222,148,52,13,1,1,15,65,240,370,558,370,240,65,15,1,1,16
%N A026648 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k) if n is even and k is odd, else T(n,k)=t(n-1,k-1)+T(n-1,k).
%F A026648 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,j)-to-(i+1,j+1) for i=0, j >= 0 if both i and j are even.
%K A026648 nonn,tabl
%O A026648 1,5
%A A026648 Clark Kimberling, ck6@cedar.evansville.edu
%I A026747
%S A026747 1,1,1,1,3,1,1,4,4,1,1,6,11,5,1,1,7,17,16,6,1,1,9,30,44,22,7,
%T A026747 1,1,10,39,74,66,29,8,1,1,12,58,143,184,95,37,9,1,1,13,70,201,
%U A026747 327,279,132,46,10,1,1,15,95,329,671,790,411,178,56,11,1,1,16
%N A026747 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k) if n is even and 1<=k<=n/2, else T(n,k)=T(n-1,k-1)+T(n-1,k).
%F A026747 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,2h+i)-to-(i+1,2h+i+1) for i >= 0, h>=0.
%K A026747 nonn,tabl
%O A026747 1,5
%A A026747 Clark Kimberling, ck6@cedar.evansville.edu
%I A026374
%S A026374 1,1,1,1,3,1,1,4,4,1,1,6,11,6,1,1,7,17,17,7,1,1,9,30,45,30,9,
%T A026374 1,1,10,39,75,75,39,10,1,1,12,58,144,195,144,58,12,1,1,13,70,
%U A026374 202,339,339,202,70,13,1,1,15,95,330,685,873,685,330,95,15,1
%N A026374 Triangular array T by rows: T(n,0)=T(n,n)=1 for all n >= 0, T(n,k)=T(n-1,k-1) for odd n and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-2,k-1) for even n and 1<=k<=n-1.
%F A026374 T(n,k) = number of integer strings s(0),...,s(n) such that s(0)=0, s(n)=n-2k, where, for 1<=i<=n,s(i) is even if i is even and |s(i)-s(i-1)|<=1.
%K A026374 nonn,tabl
%O A026374 1,5
%A A026374 Clark Kimberling, ck6@cedar.evansville.edu
%I A058879
%S A058879 1,1,1,1,1,3,1,1,4,7,1,1,4,9,18,1,1,5,10,28,44
%N A058879 Triangle T(n,k) = number of connected graphs with one cycle of length m = n-k+1 and n nodes (n >= 3, 3<=k<=n).
%D A058879 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150, Table 9.
%D A058879 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 69, (3.4.1).
%e A058879 1; 1,1; 1,1,3; 1,1,4,7; 1,1,4,9,18; ...
%Y A058879 Diagonals give A000226, A000368, A000534. Row sums give A001429.
%K A058879 nonn,easy,more,tabl,nice
%O A058879 1,6
%A A058879 njas, Jan 07 2001
%I A025255
%S A025255 1,3,1,1,4,8,22,55,145,386,1039,2837,7808,21693,60697,170959,484306,1379020,
%T A025255 3944780,11330962,32668988,94510169,274262571,798153211,2328830794,
%U A025255 6811354123,19966156857,58647721275,172600025350,508870577560,1502795261440
%N A025255 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.
%K A025255 nonn
%O A025255 1,2
%A A025255 Clark Kimberling (ck6@cedar.evansville.edu)
%I A016463
%S A016463 3,1,1,4,59,3,1,1,1,1,2,3,19,17,6,1,5,1,1,1,34,1,11,72,
%T A016463 3,1,2,1,5,2,2,2,2,14,2,16,1,1,5,11,3,1,3,4,5,2,3,1,2,1,
%U A016463 2,10,3,1,22,1,22,1,10,1,41,2,1,18,10,1,9,1,6,1,4,11,3
%N A016463 Continued fraction for ln(35).
%K A016463 nonn,cofr
%O A016463 1,1
%A A016463 njas
%I A016562
%S A016562 3,1,1,5,1,1,1,1,10,2,5,1,7,1,32,18,2,9,1,1,1,29,1,1,2,
%T A016562 50,1,29,1,1,10,1,1,3,1,1,1,2,1,1,1,3,2,3,1,2,3,1,1,6,2,
%U A016562 4,1,4,1,10,1,40,4,2,3,1,1,12,2,28,17,1,2,18,7,1,1,2,9
%N A016562 Continued fraction for ln(69/2).
%K A016562 nonn,cofr
%O A016562 1,1
%A A016562 njas
%I A046230
%S A046230 1,1,1,1,3,1,1,5,1,1,11,7,1,1,1,9,9,1,1,1,20,27,11,1,1,67,1,19,13,1,1,
%T A046230 147,105,51,15,1,1,1,126,1,78,1,33,17,1,1,1,273,1,204,1,111,83,19,1,1,
%U A046230 1,477,1,315,305,1,51,21,1,1,1,1023,1,792,935,407,123,23,1,1,1,1815
%N A046230 First denominator and then numerator of elements to right of central elements of 1/2-Pascal triangle (by row), excluding 2's.
%e A046230 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...
%Y A046230 Cf. A046213.
%K A046230 tabf,nonn
%O A046230 1,5
%A A046230 Mohammad K. Azarian, ma3@cedar.evansville.edu
%E A046230 More terms from James A. Sellers (sellersj@math.psu.edu), Dec 13 1999
%I A046229
%S A046229 1,1,1,1,3,1,1,5,1,1,11,7,1,1,9,1,9,1,1,20,1,27,11,1,1,67,19,1,13,1,1,
%T A046229 147,105,51,15,1,1,126,1,78,1,33,1,17,1,1,273,1,204,1,111,1,83,19,1,1,
%U A046229 477,1,315,1,305,51,1,21,1,1,1023,1,792,1,935,407,123,23,1,1,1815,1
%N A046229 First numerator and then denominator of elements to right of central elements of 1/2-Pascal triangle (by row), excluding 2's.
%e A046229 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...
%Y A046229 Cf. A046213.
%K A046229 nonn,tabf
%O A046229 1,5
%A A046229 Mohammad K. Azarian, ma3@cedar.evansville.edu
%E A046229 More terms from James A. Sellers (sellersj@math.psu.edu), Dec 13 1999
%I A014475
%S A014475 1,1,3,1,1,5,1,15,1,35,21,7,1,1,9,1,45,1,165,55,11,1,495,1,1287,715,13,
%T A014475 1,3003,1001,91,1,6435,5005,3003,1365,455,105,15,1,1,17,1,153,1,969,
%U A014475 171,19,1,4845,1,20349,5985,21,1,74613,7315,231,1,245157,100947,33649
%N A014475 Triangular array formed from odd elements to right of middle of rows of Pascal's triangle.
%K A014475 nonn,tabf
%O A014475 1,3
%A A014475 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A014475 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A021325
%S A021325 0,0,3,1,1,5,2,6,4,7,9,7,5,0,7,7,8,8,1,6,1,9,9,3,7,6,9,4,7,0,4,0,4,
%T A021325 9,8,4,4,2,3,6,7,6,0,1,2,4,6,1,0,5,9,1,9,0,0,3,1,1,5,2,6,4,7,9,7,5,
%U A021325 0,7,7,8,8,1,6,1,9,9,3,7,6,9,4,7,0,4,0,4,9,8,4,4,2,3,6,7,6,0,1,2,4
%N A021325 Decimal expansion of 1/321.
%K A021325 nonn,cons
%O A021325 0,3
%A A021325 njas
%I A026780
%S A026780 1,1,1,1,3,1,1,5,4,1,1,7,12,5,1,1,9,24,17,6,1,1,11,40,53,23,7,
%T A026780 1,1,13,60,117,76,30,8,1,1,15,84,217,246,106,38,9,1,1,17,112,
%U A026780 361,580,352,144,47,10,1,1,19,144,557,1158,1178,496,191,57,11
%N A026780 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k) if 1<=k<=[ n/2 ], else T(n,k)=T(n-1,k-1)+T(n-1,k).
%F A026780 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,i+h)-to-(i+1,i+h+1) for i >= 0, h>=0.
%K A026780 nonn,tabl
%O A026780 1,5
%A A026780 Clark Kimberling, ck6@cedar.evansville.edu
%I A026703
%S A026703 1,1,1,1,3,1,1,5,5,1,1,6,13,6,1,1,7,24,24,7,1,1,8,31,61,31,8,
%T A026703 1,1,9,39,116,116,39,9,1,1,10,48,155,293,155,48,10,1,1,11,58,
%U A026703 203,564,564,203,58,11,1,1,12,69,261,767,1421,767,261,69,12,1
%N A026703 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k) if k=[ n/2 ] or k=[ (n+1)/2 ], else T(n,k)=T(n-1,k-1)+T(n-1,k).
%F A026703 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,j)-to-(i+1,j+1) if |i-j|<=1.
%K A026703 nonn,tabl
%O A026703 1,5
%A A026703 Clark Kimberling, ck6@cedar.evansville.edu
%I A026615
%S A026615 1,1,1,1,3,1,1,5,5,1,1,7,10,7,1,1,9,17,17,9,1,1,11,26,34,26,11,
%T A026615 1,1,13,37,60,60,37,13,1,1,15,50,97,120,97,50,15,1,1,17,65,147,
%U A026615 217,217,147,65,17,1,1,19,82,212,364,434,364,212,82,19,1,1,21
%N A026615 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; T(n,1)=T(n,n-1)=2n-1 for n >= 1; T(n,k)=T(n-1,k-1)+T(n-1,k) for 2<=k<=n-2, n >= 4.
%F A026615 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,j)-to-(i+1,j+1) for i=0, j >= 0 and for j=0, i >= 0.
%K A026615 nonn,tabl
%O A026615 1,5
%A A026615 Clark Kimberling, ck6@cedar.evansville.edu
%I A026681
%S A026681 1,1,1,1,3,1,1,5,5,1,1,7,10,7,1,1,9,17,17,9,1,1,11,26,44,26,11,
%T A026681 1,1,13,37,87,87,37,13,1,1,15,50,150,174,150,50,15,1,1,17,65,
%U A026681 237,324,324,237,65,17,1,1,19,82,352,561,822,561,352,82,19,1
%N A026681 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-1,k) if k or n-k is of form 2h for h=1,2,...,[ n/4 ], else T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k).
%F A026681 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,j)-to-(i+1,j+1) for i even and j >= i and for j even and i >= j.
%K A026681 nonn,tabl
%O A026681 1,5
%A A026681 Clark Kimberling, ck6@cedar.evansville.edu
%I A026714
%S A026714 1,1,1,1,3,1,1,5,5,1,1,7,13,7,1,1,8,25,25,8,1,1,9,40,63,40,9,
%T A026714 1,1,10,49,128,128,49,10,1,1,11,59,217,319,217,59,11,1,1,12,70,
%U A026714 276,664,664,276,70,12,1,1,13,82,346,1157,1647,1157,346,82,13
%N A026714 Triangular array T by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k) if k=[ (n-1)/2 ] or k=[ n/2 ] or k=[ (n+2)/2 ], else T(n,k)=T(n-1,k-1)+T(n-1,k).
%F A026714 T(n,k) = number of paths from (0,0) to (n-k,k) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j >= 0 and edges (i,j)-to-(i+1,j+1) if |i-j|<=2.
%K A026714 nonn,tabl
%O A026714 1,5
%A A026714 Clark Kimberling, ck6@cedar.evansville.edu
%I A008288
%S A008288 1,1,1,1,3,1,1,5,5,1,1,7,13,7,1,1,9,25,25,9,1,1,11,41,63,41,11,1,1,13,61,
%T A008288 129,129,61,13,1,1,15,85,231,321,231,85,15,1,1,17,113,377,681,681,377,113,
%U A008288 17,1,1,19,145,575,1289,1683,1289,575,145,19,1,1,21,181,833,2241,3653,3653
%N A008288 Table of Delannoy numbers T(n,k) (n >= 0, k >= 0) read by antidiagonals.
%D A008288 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 37.
%D A008288 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A008288 H. Delannoy. Emploi de l'echiquier pour la resolution de certains problemes de probabilites, Association Francaise pour l'Avancement des Sciences, 18-th session, 1895. pp.70-90 (table given pp. 76)
%D A008288 E. Lucas. Theorie des Nombres. Gauthier-Villard 1891, pp.174
%D A008288 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 6.3.8.
%D A008288 R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
%F A008288 T(n,0) = 1 = T(n,n) for n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for k = 1,2,...,n-1, for n >= 2.
%F A008288 Sum_{n >= 0, k >= 0} T(n,k)*x^n*y^k = 1/(1-x-y-x*y).
%F A008288 T(n,k) = Sum_{d} binomial(k,d)*binomial(n+k-d,k) = Sum_{d} 2^d*binomial(n,d)*binomial(k,d).
%e A008288 Table begins:
%e A008288 1 1 1 1 1 1 1 1
%e A008288 1 3 5 7 9 11 13 ...
%e A008288 1 5 13 25 41 61 ...
%e A008288 1 7 25 63 129 231 ...
%e A008288 1 9 41 129 321 681 ...
%p A008288 A008288:=proc(n,k) option remember; if n = 1 then 1; elif k = 1 then 1; else A008288(n-1,k-1)+A008288(n,k-1)+A008288(n-1,k); fi; end;
%p A008288 read transforms; SERIES2(1/(1-x-y-x*y),x,y,12): SERIES2TOLIST(%,x,y,12);
%Y A008288 Sums of antidiagonals = A000129 (Pell numbers), T(n, n) = A001850 (Delannoy numbers), (T(n, 3)) = A001845, (T(n, 4)) = A001846, etc. See also A027618. Rows and diagonals give A001844-A001850. Cf. A059446.
%Y A008288 Has same main diagonal as A064861.
%K A008288 nonn,tabl,nice,easy
%O A008288 0,5
%A A008288 njas
%E A008288 Expanded description from Clark Kimberling 6/97. Additional references from Sylviane R. Schwer (schwer@lipn.univ-paris13.fr), Nov 28 2001.
%I A056152
%S A056152 1,1,1,1,3,1,1,5,5,1,1,8,17,8,1,1,11,42,42,11,1,1,15,91,179,91,15,1,1,
%T A056152 19,180,633,633,180,19,1,1,24,328,2001,3835,2001,328,24,1,1,29,565,
%U A056152 5745,20755,20755,5745,565,29,1,1,35,930,15274,102089,200082,102089
%N A056152 Trianguler array giving number of bipartite graphs with n vertices, no isolated vertices, and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.
%e A056152 [1], [1, 1], [1, 3, 1], [1, 5, 5, 1], [1, 8, 17, 8, 1], ...; There are 17 bipartite graphs with 6 vertices, no isolated vertices, and a distinguished bipartite block with 3 vertices, or equivalently, there are 17 3x3 binary matrices with no zero rows or columns, up to row and column permutation:
%e A056152 [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
%e A056152 [0 0 1] [0 0 1] [0 1 0] [0 1 0] [0 1 0] [0 1 1] [0 1 1] [0 1 1] [1 1 0]
%e A056152 [1 1 0] [1 1 1] [1 0 0] [1 0 1] [1 1 1] [1 0 1] [1 1 0] [1 1 1] [1 1 0]
%e A056152
%e A056152 [0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
%e A056152 [1 1 0] [1 1 1] [0 1 1] [0 1 1] [1 0 1] [1 0 1] [1 1 1] [1 1 1]
%e A056152 [1 1 1] [1 1 1] [1 0 1] [1 1 1] [1 1 0] [1 1 1] [1 1 1] [1 1 1].
%Y A056152 Row sums give A055192.
%Y A056152 Cf. A049312, A048194, A028657, A049311, A024206, A055609, A055082-A055084.
%K A056152 nonn
%O A056152 0,5
%A A056152 Vladeta Jovovic (vladeta@Eunet.yu), Jul 29 2000
%I A054142
%S A054142 1,1,1,1,3,1,1,5,6,1,1,7,15,10,1,1,9,28,35,15,1,1,11,45,84,70,21,1,1,
%T A054142 13,66,165,210,126,28,1,1,15,91,286,495,462,210,36,1,1,17,120,455,1001,
%U A054142 1287,924,330,45,1,1,19,153,680,1820,3003
%N A054142 Triangular array C(2n-k,k), k=0,1,...,n, n >= 0.
%C A054142 Row sums are odd-indexed Fibonacci numbers.
%e A054142 Rows: {1}, {1,1}, {1,3,1}, {1,5,6,1}, ...
%Y A054142 These are the even-indexed rows of A011973; the odd-indexed rows form A053123.
%K A054142 nonn
%O A054142 1,5
%A A054142 Clark Kimberling, ck6@cedar.evansville.edu
%I A047969
%S A047969 1,1,1,1,3,1,1,5,7,1,1,7,19,15,1,1,9,37,65,31,1,1,11,61,175,211,
%T A047969 63,1,1,13,91,369,781,665,127,1,1,15,127,671,2101,3367,2059,255,
%U A047969 1,1,17,169,1105,4651,11529,14197,6305,511,1,1,19,217,1695,9031
%N A047969 Square array a(n,k)=(n+1)^(k+1)-n^(k+1) (n >= 0, k >= 0) read by antidiagonals.
%C A047969 If each row started with an initial 0 (i.e. a(n,k)=(n+1)^k-n^k) then each row would be the binomial transform of the preceding row. - Henry Bottomley (se16@btinternet.com), May 31 2001
%e A047969 Array begins
%e A047969 1 1 1 1 1 1 1 1 ...
%e A047969 1 3 7 15 31 63 ...
%e A047969 1 5 19 65 211 ...
%e A047969 1 7 37 175 ...
%Y A047969 Cf. A047970.
%K A047969 nonn,tabl,nice,easy
%O A047969 0,5
%A A047969 njas
%I A047812
%S A047812 1,1,1,1,3,1,1,5,7,1,1,9,20,11,1,1,13,48,51,18,1,1,20,100,169,112,26,1,
%T A047812 1,28,194,461,486,221,38,1,1,40,352,1128,1667,1210,411,52,1,1,54,615,
%U A047812 2517,4959,5095,2761,720,73,1,1,75,1034,5288,13241,18084,13894,5850
%N A047812 Parker's partition triangle read by rows.
%C A047812 The entries in row n are the coefficients of q^(k(n+1)) in the q-binomial coefficient [2n,n] where k runs from 0 to n-1 - James A. Sellers (sellersj@math.psu.edu).
%D A047812 R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.
%e A047812 1; 1 1; 1 3 1; 1 5 7 1; ...
%K A047812 nonn,tabl,easy,nice
%O A047812 0,5
%A A047812 njas
%E A047812 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A061857
%S A061857 0,1,0,3,1,1,6,2,2,1,10,4,4,2,2,15,6,5,3,3,2,21,9,7,5,4,3,3,28,12,10,6,
%T A061857 6,4,4,3,36,16,12,8,8,5,5,4,4,45,20,15,10,9,7,6,5,5,4,55,25,19,13,11,9,
%U A061857 8,6,6,5,5,66,30,22,15,13,10,10,7,7,6,6,5,78,36,26,18,16,12,12,9,8,7,7
%N A061857 Triangle where the k-th item at n-th row (both starting from 1) tells in how many ways we can add 2 distinct integers from 1 to n, in such way that the sum is divisible by k.
%e A061857 E.g. the second term on the sixth row is 6, because we have 6 solutions: {1+3, 1+5, 2+4, 2+6, 3+5, 4+6}, and the third term on the same row is 5, because we have solutions {1+2,1+5,2+4,3+6,4+5}
%p A061857 [seq(DivSumChoose2Triangle(j),j=1..120)]; DivSumChoose2Triangle := (n) -> nops(DivSumChoose2(trinv(n-1),(n-((trinv(n-1)*(trinv(n-1)-1))/2))));
%p A061857 DivSumChoose2 := proc(n,k) local a,i,j; a := []; for i from 1 to (n-1) do for j from (i+1) to n do if(0 = ((i+j) mod k)) then a := [op(a),[i,j]]; fi; od; od; RETURN(a); end;
%Y A061857 The left edge (first diagonal) of the triangle: A000217, the second diagonal is given by C(((n+(n mod 2))/2), 2)+C(((n-(n mod 2))/2), 2) = A002620, the third diagonal by A058212, the fourth by A056837, the central column by A042963 ??? trinv given at A054425. Cf. A061865.
%K A061857 nonn,tabl
%O A061857 0,4
%A A061857 Antti.Karttunen@iki.fi May 11 2001
%I A055898
%S A055898 1,1,1,1,3,1,1,6,5,1,1,10,16,7,1,1,15,39,31,9,1,1,21,81,101,51,11,1,1,
%T A055898 28,150,272,209,76,13,1,1,36,256,636,696,376,106,15,1,1,45,410,1340,
%U A055898 1980,1496,615,141,17,1,1,55,625,2600,5000,5032,2850,939,181,19,1,1
%N A055898 Triangle: Number of directed site animals on a square lattice with n+1 total sites and k sites supported in one particular way.
%D A055898 M. Bousquet-M\'{e}lou, New enumerative results on two-dimensional directed animals, Discr. Math., 180 (1998), 73-106.
%H A055898 A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189 (A_sq of Section 6).
%F A055898 G.f.: A(x,y)=(1/2x)((1-(4x/((1+x)(1+x-xy))))^(-1/2) - 1).
%e A055898 1; 1,1; 1,3,1; 1,6,5,1; 1,10,16,7,1; ...
%Y A055898 Row sums give A005773. Columns 0-8: A000012, A000217, A011863(n-1), A055899-A055904. Cf. A055905, A055907.
%K A055898 nonn,tabl
%O A055898 0,5
%A A055898 Christian G. Bower (bowerc@usa.net), Jun 13 2000
%I A035582
%S A035582 0,0,0,0,0,0,0,0,1,0,0,1,1,3,1,1,6,5,7,6,8,18,16,18,24,29,48,44,50,72,83,
%T A035582 113,117,131,187,209,261,285,325,440,491,576,662,755,978,1084,1245,1458,
%U A035582 1671,2068,2304,2609,3096,3526,4232,4730,5348,6324,7187,8410,9462,10665
%N A035582 Partitions of n with equal nonzero number of parts congruent to each of 0, 1 and 2 (mod 5)
%K A035582 nonn,part
%O A035582 0,14
%A A035582 Olivier Gerard (ogerard@ext.jussieu.fr)
%E A035582 More terms from dww
%I A054120
%S A054120 1,1,1,1,3,1,1,6,6,1,1,9,18,9,1,1,12,39,39,12,1,1,15,69,114,69,15,1,1,
%T A054120 18,108,261,261,108,18,1,1,21,156,507,750,507,156,21,1,1,24,213,879,
%U A054120 1779,1779,879,213,24,1,1,27,279,1404,3672,5058
%N A054120 Triangular array T(n,k): start with T(n,0)=T(n,n)=1 for n >= 0; recursively, draw vertical lines through T(n-1,k-1) if present and T(n-1,k) if present; then T(n,k) is the sum of T(i,j) that lie on or between the lines and not below T(n,k).
%e A054120 Rows: {1}, {1,1}, {1,3,1}, {1,6,6,1}, ...
%Y A054120 Row sums: A052945.
%K A054120 nonn,tabl,eigen
%O A054120 0,5
%A A054120 Clark Kimberling, ck6@cedar.evansville.edu
%I A056241
%S A056241 1,1,1,1,3,1,1,6,6,1,1,10,19,10,1,1,15,45,45,15,1,1,21,90,141,90,21,1,
%T A056241 1,28,161,357,357,161,28,1,1,36,266,784,1107,784,266,36,1,1,45,414,
%U A056241 1554,2907,2907,1554,414,45,1,1,55,615,2850,6765,8953,6765,2850,615,55
%N A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).
%D A056241 Hwang, F. K.; Mallows, C. L.; Enumerating nested and consecutive partitions. J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%F A056241 T(n,k) = Sum_{j=0..k-1} C(n-1,2k-j-2)*C(2k-j-2,j).
%e A056241 1; 1,1; 1,3,1; 1,6,6,1; 1,10,19,10,1; ...
%Y A056241 Columns are A000217, A005712, A005714, A005716.
%K A056241 nonn,tabl,easy,nice
%O A056241 1,5
%A A056241 Colin L. Mallows (colinm@research.avayalabs.com), Aug 23 2000
%E A056241 More terms from James A. Sellers (sellersj@math.psu.edu), Aug 25 2000
%I A001263
%S A001263 1,1,1,1,3,1,1,6,6,1,1,10,20,10,1,1,15,50,50,15,1,1,21,105,175,105,
%T A001263 21,1,1,28,196,490,490,196,28,1,1,36,336,1176,1764,1176,336,36,1,
%U A001263 1,45,540,2520,5292,5292,2520,540,45,1,1,55,825,4950,13860,19404
%N A001263 Triangle of Narayana numbers C(n,k)C(n,k-1)/n. Also called the Catalan triangle.
%C A001263 Antichains (or order ideals) in the poset 2*(k-1)*(n-k) or plane partitions with rows <= k-1, columns <= n-k, and entries <= 2 - Mitch Harris (maharri@cs.uiuc.edu), July 2000
%D A001263 Benchekroun, S.; Moszkowski, P.; A bijective proof of an enumerative property of legal bracketings. Discrete Math. 176 (1997), no. 1-3, 273-277.
%D A001263 Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p.103-124
%D A001263 Hwang, F. K.; Mallows, C. L.; Enumerating nested and consecutive partitions. J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%D A001263 G. Kreweras, Sur les e'ventails de segments, {\em Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle}, Institut de Statistique, Universit\'{e} de Paris, #15 (1970), 3-41.
%D A001263 P. A. MacMahon, Combinatory Analysis, Sect. 495, 1916.
%D A001263 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
%D A001263 R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p.259-279. Thm. 18.1
%D A001263 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.36(a).
%e A001263 1; 1,1; 1,3,1; 1,6,6,1; 1,10,20,10,1; 1,15,50,50,15,1; ...
%p A001263 f:=(n,k)->binomial(n,k)*binomial(n,k+1)/n;
%Y A001263 Cf. A000372, A002083, A056932, A056939, A056940, A056941.
%Y A001263 Columns give A000217, A002415 (or A008057), A006542, A006857.
%K A001263 nonn,easy,tabl,nice
%O A001263 1,5
%A A001263 njas
%I A008278
%S A008278 1,1,1,1,3,1,1,6,7,1,1,10,25,15,1,1,15,65,90,31,1,1,21,140,
%T A008278 350,301,63,1,1,28,266,1050,1701,966,127,1,1,36,462,2646,
%U A008278 6951,7770,3025,255,1,1,45,750,5880,22827,42525,34105,9330
%N A008278 Triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1<=k<=n.
%D A008278 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
%D A008278 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%Y A008278 See A008277 and A048993, which are the main entries for this triangle of numbers.
%K A008278 nonn,tabl,nice
%O A008278 1,5
%A A008278 njas
%I A056858
%S A056858 1,1,1,1,3,1,1,6,7,1,1,10,26,14,1,1,15,71,89,26,1
%N A056858 Triangle of number of rises in set partitions of n.
%C A056858 Number of rises s_{i+1} > s_i in a set partition {s_1, ..., s_n} of {1, ..., n}, where s_i is the subset containing i, s(1) = 1, and s(i) <= 1 + max of previous s(j)'s.
%D A056858 W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.
%e A056858 For example {1, 2, 1, 2, 2, 3} is a set partition of {1, 2, 3, 4, 5, 6} and has 3 rises, at i = 1, i = 3, and i = 5.
%e A056858 1; 1,1; 1,3,1; 1,6,7,1; 1,10,26,14,1; 1,15,71,89,26,1; ...
%Y A056858 Cf. Bell numbers A000110.
%Y A056858 Cf. A056857-A056863.
%K A056858 easy,nonn,tabl,more
%O A056858 1,5
%A A056858 Winston C. Yang (winston@cs.wisc.edu), Aug 31 2000
%I A046716
%S A046716 1,1,1,1,3,1,1,6,8,1,1,10,29,24,1,1,15,75,145,89,1,1,21,160,545,814,
%T A046716 415,1
%N A046716 Coefficients of a special case of Poisson-Charlier polynomials.
%D A046716 E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
%F A046716 Reference gives a recurrence.
%e A046716 1; 1,1; 1,3,1; 1,6,8,1; 1,10,29,24,1; ...
%K A046716 nonn,tabl,easy,more
%O A046716 0,5
%A A046716 njas
%I A053193
%S A053193 1,1,1,3,1,1,7,1,1,9,1,5,9,1,1,13,11,1,15,1,1,21,1,7,19,1,15,21,1,1,27,
%T A053193 17,1,25,1,1,35,17,1,27,1,21,31,1,19,33,23,1,39,1,1,57,1,1,39,1,27,45,
%U A053193 23,11,43,25,1,45,1,25,63,1,1,49,23,33,63,1,1,57,35,1,55,29,1,85,1,13
%N A053193 Cototient of odd numbers.
%F A053193 a(n)=cototient[2n+1]=A051953(2n+1)
%e A053193 n=12, 2n+1=25, Phi(25)=20, Cototient(25)=25-20=5, a(12)=5 n=16, 2n+1=33, Phi(33)=20, Cototient(33)=33-20=13, a(16)=13
%Y A053193 A051953.
%K A053193 nonn
%O A053193 1,4
%A A053193 Labos E. (labos@ana1.sote.hu), Mar 02 2000
%I A010273
%S A010273 3,1,1,7,1,2,1,4,3,12,1,1,1,1,3,4,1,4,1,2,2,1,17,2,18,2,
%T A010273 16,1,2,45,1,2,6,2,3,111,16,2,1,1,3,31,1,4,1,1,1,12,1,19,
%U A010273 1,8,3,1,1,1,1,1,2,1,2,1,8,2,1,3,7,2,20,5,2,25,1,3,22,2
%N A010273 Continued fraction for cube root of 44.
%K A010273 nonn,cofr
%O A010273 0,1
%A A010273 njas
%I A046143
%S A046143 1,1,3,1,1,7,1,3,1,15,1,1,1,1,31,1,3,7,3,1,63,1,1,1,1,1,1,127,1,3,1,15,
%T A046143 1,3,1,255,1,1,7,1,1,7,1,1,511,1,3,1,3,31,3,1,3,1,1023,1,1,1,1,1,1,1,1,
%U A046143 1,1,2047,1,3,7,15,1,63,1,15,7,3,1,4095,1,1,1,1,1,1,1,1,1,1,1,1
%N A046143 Triangle of GCD[ 2^p-1,2^q-1 ] = 2^GCD[ p,q ]-1.
%H A046143 E. W. Weisstein, More information
%K A046143 nonn,tabl
%O A046143 1,3
%A A046143 Eric W. Weisstein (eric@weisstein.com)
%I A065625
%S A065625 3,1,1,7,5,1,2,3,2,1,6,2,7,2,1,14,11,4,3,2,1,15,6,5,9,3,2,1,4,7,3,5,4,3,2,1,5,4,15,6,
%T A065625 11,4,3,2,1,12,10,8,7,6,5,4,3,2,1,13,22,9,4,7,13,5,4,3,2,1,28,23,10,19,8,7,6,5,4,3,2,1,29,12,
%U A065625 11,10,9,8,15,6,5,4,3,2,1,30,13,6,11,5,9,8,7,6,5,4,3,2,1,31,14,14,12,23,10,9,17,7,6,5,4,3,2
%N A065625 Table of permutations of N, each row being a generator of the "rotation group" of infinite planar binary tree. Inverse generators are given in A065626.
%C A065625 Consider the following infinite binary tree, where the nodes are numbered in breadth-first, left-to-right fashion from the top as:
%C A065625 .............................1............................
%C A065625 .............2...............................3............
%C A065625 .....4...............5...............6...............7....
%C A065625 .8.......9.......10.....11.......12.....13.......14.....15
%C A065625 etc, i.e. the node Y is a descendant of the node X, iff its binary expansion (the most significant bits) begin with the binary expansion of X.
%C A065625 In this table the nth row is a permutation induced by the rotation of the node n right, and in the table A065626 the corresponding row gives the inverse of that permutation, induced by rotation of the node n left. Particular realizations of this tree are the Christoffel tree, and the Stern-Brocot tree (A007305/A007306), thus each such rotation, or composition of such rotations (e.g. A065249) induces a particular bijective function on rationals, and such functions form the "group A" of the order preserving permutations of the rational numbers as defined by Cameron.
%H A065625 A. Karttunen: How to generate A065249 and A065250
%p A065625 [seq(RotateRightTable(j),j=0..119)];
%p A065625 RotateRightTable := n -> RotateNodeRight(1+(n-((trinv(n)*(trinv(n)-1))/2)),(((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1);
%p A065625 # Rewrites t-prefixed x's in the following way: t -> t1, t1... -> t11..., t0 -> t, t01... -> t10..., t00... -> t0..., and leaves other x's intact.
%p A065625 RotateNodeRight := proc(t,x) local u,y; u := floor_log_2(t)+1; y := floor_log_2(x)+1; if(y < u) then RETURN(x); fi; if(floor(x/(2^(y-u))) <> t) then RETURN(x); fi; if(x = t) then RETURN((2*x)+1); fi; if(1 = (floor(x/(2^(y-u-1))) mod 2)) then RETURN(x + (t * 2^(y-u)) + 2^(y-u)); fi; if(y = (u+1)) then RETURN(x/2); fi; if(1 = (floor(x/(2^(y-u-2))) mod 2)) then RETURN(x + 2^(y-u-2)); fi; RETURN(x - (t * 2^(y-u-1))); end;
%Y A065625 The first row (rotate the top node right): A057114, 2nd row (rotate the top node's left child): A065627, 3rd row (rotate the top node's right child): A065629, 4th row: A065631, 5th row: A065633, 6th row: A065635, 7th row: A065637, 8th row: A065639. Maple procedure floor_log_2 given in A054429, for trinv, follow A065167.
%Y A065625 Variant of the same idea: A065658.
%K A065625 nonn,tabl
%O A065625 0,1
%A A065625 Antti.Karttunen@iki.fi Nov 8 2001
%I A008277
%S A008277 1,1,1,1,3,1,1,7,6,1,1,15,25,10,1,1,31,90,65,15,1,1,63,301,
%T A008277 350,140,21,1,1,127,966,1701,1050,266,28,1,1,255,3025,7770,
%U A008277 6951,2646,462,36,1,1,511,9330,34105,42525,22827,5880,750,45,1
%N A008277 Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
%C A008277 Also known as Stirling set numbers. S2(n,k) enumerates partitions of an n-set into k non-empty subsets.
%D A008277 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
%D A008277 Bleick, W. E.; Wang, Peter C. C. Asymptotics of Stirling numbers of the second kind. Proc. Amer. Math. Soc. 42 (1974), 575-580.
%D A008277 Bleick, W. E.; Wang, Peter C. C. Erratum to: "Asymptotics of Stirling numbers of the second kind" (Proc. Amer. Math. Soc. {42} (1974), 575-580). Proc. Amer. Math. Soc. 48 (1975), 518.
%D A008277 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 42.
%D A008277 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.
%D A008277 J. H. Conway and Richard K. Guy, The Book of Numbers, Springer, p. 92.
%D A008277 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%D A008277 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 244.
%D A008277 Knessl, Charles; Keller, Joseph B. Stirling number asymptotics from recursion equations using the ray method. Stud. Appl. Math. 84 (1991), no. 1, 43-56.
%D A008277 Korshunov, A. D. Asymptotic behavior of Stirling numbers of the second kind. (Russian) Metody Diskret. Analiz. No. 39 (1983), 24-41.
%D A008277 J. Riordan, An Introduction to Combinatorial Analysis, p. 48.
%D A008277 Temme, N. M. Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (1993), no. 3, 233-243.
%D A008277 Timashev, A. N. On asymptotic expansions of Stirling numbers of the first and second kinds. (Russian) Diskret. Mat. 10 (1998), no. 3,148-159 translation in Discrete Math. Appl. 8 (1998), no. 5, 533-544.
%D A008277 M. C. Wolf, Symmetric functions for non-commutative elements, Duke Math. J., 2 (1936), 626-637.
%H A008277 R. M. Dickau, Stirling numbers of the second kind
%H A008277 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A008277 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A008277 S2(n,k) = k*S2(n-1,k)+S2(n-1,k-1), n>1. S2(1,k) = 0, k>1. S2(1,1)=1.
%F A008277 E.g.f.: A(x,y)=exp(y*exp(x)-y). E.g.f. m-th column: ((exp(x)-1)^m)/m!.
%F A008277 S2(n,k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i)*C(k,i)*i^n.
%e A008277 1; 1,1; 1,3,1; 1,7,6,1; 1,15,25,10,1; ...
%o A008277 (PARI) S2(n,k) = if(k<1|k>n,0,if(n==1,1,k*S2(n-1,k)+S2(n-1,k-1))); printp(matrix(9,9,n,k,S2(n,k)))
%Y A008277 Cf. A008275, A039810-A039813, A048993, A048994, A028246.
%Y A008277 A000110(n) = sum(S(n, k)) k=1..n, n>0.
%K A008277 nonn,tabl,nice
%O A008277 1,5
%A A008277 njas
%I A063394
%S A063394 1,1,1,1,3,1,1,7,7,1,1,15,19,15,1,1,31,47,47,31,1,1,63,111,131,111,63,
%T A063394 1,1,127,255,343,343,255,127,1,1,255,575,863,979,863,575,255,1,1,511,
%U A063394 1279,2111,2655,2655,211,1279,511,1,1,1023,2815,5055,6943,7683,6943
%N A063394 Border sum triangle: Let T(n,0)=T(0,n)=1. T(n,m) is the sum of the elements (apart from T(n,m) itself) in the border of the rectangle with vertices T(0,0), T(n,0), T(n,m) and T(0,m).
%F A063394 T(n,m) = SUM{0 < i < n}[T(i,0)+T(i,m)] + SUM{0 <= i < m}[T(0,i)+T(n,i)] + T(0,m).
%e A063394 The triangle begins:
%e A063394 ..........1
%e A063394 ........1...1
%e A063394 ......1...3...1
%e A063394 ....1...7...7...1
%e A063394 ..1..15..19...17..1
%Y A063394 T(1, n) gives A000225(n+1), T(2, n) for n>0 gives A006589.
%K A063394 easy,nonn,tabl
%O A063394 0,5
%A A063394 Floor van Lamoen (f.v.lamoen@wxs.nl), Jul 16 2001
%I A046802
%S A046802 1,1,1,1,3,1,1,7,7,1,1,15,33,15,1,1,31,131,131,31,1,1,63,473,883,
%T A046802 473,63,1
%N A046802 Triangle of numbers related to Eulerian numbers.
%D A046802 D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
%F A046802 a(m,n) = Sum C(m-1,n-1)*L(m-r,n-1), L = Eulerian numbers.
%e A046802 1; 1 1; 1 3 1; 1 7 7 1; ...
%Y A046802 Cf. A008292. Row sums give A000522.
%K A046802 nonn,tabl,easy,nice,more
%O A046802 1,5
%A A046802 njas
%I A022166
%S A022166 1,1,1,1,3,1,1,7,7,1,1,15,35,15,1,1,31,155,155,31,1,1,63,
%T A022166 651,1395,651,63,1,1,127,2667,11811,11811,2667,127,1,1,
%U A022166 255,10795,97155,200787,97155,10795,255,1,1,511,43435,788035
%N A022166 Triangle of Gaussian binomial coefficients [ n,k ] for q = 2.
%D A022166 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%K A022166 nonn,tabl
%O A022166 0,5
%A A022166 njas
%I A058669
%S A058669 1,1,1,1,3,1,1,7,7,1,1,15,36,15,1,1,31,171,171,31,1,1,63,813,2053,
%T A058669 813,63,1,1,127,4012,33442,33442,4012,127,1,1,255,20891,1022217,8520812,
%U A058669 1022217,20891,255,1,1,511
%N A058669 Triangle T(n,k) giving number of matroids of rank k on n labeled points (n >= 0, 0<=k<=n).
%H A058669 Index entries for sequences related to matroids
%H A058669 W. M. B. Dukes, Tables of matroids
%H A058669 W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
%e A058669 1; 1,1; 1,3,1; 1,7,7,1; ...
%Y A058669 Row sums give A058673. Diagonals give A058681, A058687. Cf. A053534.
%K A058669 nonn,nice,tabl
%O A058669 0,5
%A A058669 njas, Dec 30 2000
%I A057004
%S A057004 1,1,1,1,3,1,1,7,7,1,1,15,41,26,1,1,31,235,604,97,1,1,63,1361,14120,
%T A057004 13753,624,1,1,127,7987,334576,1712845,504243,4163,1,1,255,47321,
%U A057004 7987616,207009649,371515454,24824785,34470,1,1,511,281995,191318464
%N A057004 Array T(n,k) = number of conjugacy classes of subgroups of index k in free group of rank n, read by antidiagonals.
%D A057004 J. H. Kwak and J. Lee, J. Graph Th., 23 (1996), 105-109.
%D A057004 V. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
%D A057004 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.
%H A057004 J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
%e A057004 Array T(n,k) begins:
%e A057004 1 1 1 1 1 1 1 ...
%e A057004 1 3 7 26 97 624 4163 ...
%e A057004 1 7 41 604 13753 504243 ...
%e A057004 1 15 235 14120 1712845 ...
%Y A057004 Rows, columns, main diagonal give A057005-A057013.
%K A057004 nonn,tabl,nice,new
%O A057004 1,5
%A A057004 njas, Sep 09 2000
%E A057004 More terms from Francisco Salinas (franciscodesalinas@hotmail.com), Dec 25 2001
%I A059328
%S A059328 1,1,1,1,3,1,1,7,7,1,1,15,63,15,1,1,31,1023,1023,31,1,1,63,32767,
%T A059328 1048575,32767,63,1,1,127,2097151,34359738367,34359738367,2097151,127,
%U A059328 1,1,255,268435455,72057594037927935,1180591620717411303423
%N A059328 Table T(n,k) = T(n - 1,k) + T(n,k - 1) + T(n - 1,k)*T(n,k - 1) starting with T(0,0)=1, read by antidiagonals.
%F A059328 T(n,k)=2^C(n+k,n)-1; a(n)=2^A007318(n)-1.
%K A059328 nonn,tabl
%O A059328 0,5
%A A059328 Henry Bottomley (se16@btinternet.com), Jan 26 2001
%I A049290
%S A049290 1,1,1,1,3,1,1,7,13,1,1,15,97,71,1,1,31,625,2143,461,1
%N A049290 Array T(n,k) = number of subgroups of index k in free group of rank n, read by antidiagonals.
%D A049290 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
%D A049290 V. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
%D A049290 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).
%H A049290 J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
%e A049290 Array T(n,k) (n >= 1, k >= 1) begins:
%e A049290 1,1,1,1,1,1,1,...
%e A049290 1,3,13,71,461,...
%e A049290 1,7,97,2143,...
%e A049290 1,15,625,54335,...
%Y A049290 Rows give A003319, A027837, A049291; columns give A049294, A049295. main diagonal is A057014.
%K A049290 nonn,easy,nice,tabl,more
%O A049290 1,5
%A A049290 njas, Sep 09 2000
%I A016462
%S A016462 3,1,1,8,1,61,36,1,6,1,5,1,116,1,5,1,6,1,4,1,1,2,1,1,7,
%T A016462 68,5,1,2,1,5,1,2,1,2,17,1,2,4,1,1,1,4,2,9,5,3,11,1,3,1,
%U A016462 4,2,1,1,2,2,7,1,1,1,3,33,1,10,5,2,1,3,1,1,135,1,1,1,2
%N A016462 Continued fraction for ln(34).
%K A016462 nonn,cofr
%O A016462 1,1
%A A016462 njas
%I A034801
%S A034801 1,1,1,1,3,1,1,8,8,1,1,21,56,21,1,1,55,385,385,55,1,1,144,2640,6930,
%T A034801 2640,144,1,1,377,18096,124410,124410,18096,377,1,1,987,124033,2232594,
%U A034801 5847270,2232594,124033,987,1,1,2584,850136,40062659,274715376
%N A034801 Triangle of Fibonomial coefficients (k=2).
%D A034801 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 88.
%F A034801 Fibonomial coefficients formed from sequence F_3k [ 2,8,34,... ].
%F A034801 a(n,k) = product(fibonacci(2*(n-j)), j=0..k-1)/product(fibonacci(2*j), j=1..k)
%Y A034801 Cf. A010048.
%K A034801 nonn,tabl
%O A034801 0,5
%A A034801 njas
%E A034801 More terms from James A. Sellers (sellersj@math.psu.edu), Feb 09 2000
%I A050153
%S A050153 0,1,0,3,1,1,9,5,6,1,28,20,27,8,1,90,75,110,44,10,1,297,275,429,208,65,
%T A050153 12,1,1001,1001,1638,910,350,90,14,1,3432,3640,6188,3808,1700,544,119,
%U A050153 16,1
%N A050153 T(n,k)=M(2n+1,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.
%e A050153 Rows: {0}; {1,0}; {3,1,1}; ...
%K A050153 nonn,tabl
%O A050153 0,4
%A A050153 Clark Kimberling, ck6@cedar.evansville.edu
%I A045912
%S A045912 1,1,1,1,3,1,1,9,9,1,1,29,72,29,1,1,99,626,626,99,1,1,351,6084,
%T A045912 13869,6084,351,1,1,1275,64974,347020,347020,64974,1275,1,1,4707,
%U A045912 744193,9952274,21537270,9952274,744193,4707,1,1,17577,8965323
%N A045912 Triangle of (absolute values of) coefficients of characteristic polynomial of Pascal matrix with (i,j)-th entry C(i+j-2,i-1).
%C A045912 Sum of k-th row is A006366(n). Columns give A006134, A006135, A006136.
%D A045912 W. F. Lunnon "The Pascal matrix", Fib. Quart. vol. 15 (1977) pp 201-204.
%e A045912 1; 1,1; 1,3,1; 1,9,9,1; 1,29,72,29,1;...
%K A045912 nonn
%O A045912 0,5
%A A045912 fred@csa5.cs.may.ie (Fred Lunnon)
%I A060543
%S A060543 1,1,1,1,3,1,1,10,5,1,1,35,28,7,1,1,126,165,55,9,1,1,462,1001,455,91,
%T A060543 11,1,1,1716,6188,3876,969,136,13,1,1,6435,38760,33649,10626,1771,190,
%U A060543 15,1,1,24310,245157,296010,118755,23751,2925,253,17,1,1,92378,1562275
%N A060543 Square array by anti-diagonals of C(nk,k)/n.
%F A060543 a(n) =A060539(n,k)/n =A007318(nk,k)/n =A060540(n,k)/A060540(n-1,k)
%Y A060543 Rows include A000012, A001700, A025174. Columns include A000012, A005408, A060544. Main diagonal is A060545.
%K A060543 nonn,tabl
%O A060543 1,5
%A A060543 Henry Bottomley (se16@btinternet.com), Apr 02 2001
%I A060540
%S A060540 1,1,1,1,3,1,1,10,15,1,1,35,280,105,1,1,126,5775,15400,945,1,1,462,
%T A060540 126126,2627625,1401400,10395,1,1,1716,2858856,488864376,2546168625,
%U A060540 190590400,135135,1,1,6435,66512160,96197645544,5194672859376
%N A060540 Square array by anti-diagonals of number of ways of dividing nk labeled items into n unlabeled boxes with k items in each box.
%F A060540 T(n,k) =(nk)!/(k!^n*n!) =T(n-1,k)*A060543(n,k) =A060538(n,k)/A000142(k).
%Y A060540 Rows include A000012, A001700, A060542. Rows include A000012, A001147, A025035, A025036, etc. Main diagonal is A057599.
%K A060540 nonn,tabl
%O A060540 1,5
%A A060540 Henry Bottomley (se16@btinternet.com), Apr 02 2001
%I A013561
%S A013561 1,1,1,1,3,1,1,11,11,1,1,31,90,31,1,1,85,544,544,85,1,1,225,2997,6559,2997,225,1
%N A013561 Triangle of coefficients of polynomials arising as numerators of certain Hilbert series.
%D A013561 D.-N. Verma (dnverma@math.tifr.res.in), Towards Classifying Finite Point-Set Configurations, preprint, 1997.
%K A013561 tabl,nonn
%O A013561 3,5
%A A013561 njas
%I A055154
%S A055154 1,1,3,1,1,12,32,35,21,7,1,1,39,321,1225,2919,4977,6431,6435,5005,3003,
%T A055154 1365,455,105,15,1,1,120,2560,24990,155106,711326,2597410,7856550,
%U A055154 20135050,44337150,84665490,141118250,206252550,265182450,300540190
%N A055154 Triangle T(n,k) of k-covers of a labeled n-set, k=1..2^n-1.
%C A055154 Row sums give A003465.
%F A055154 T(n,k)=Sum_{j=0..n} (-1)^j*C(n,j)*C(2^(n-j)-1,k), k=1..2^n-1.
%e A055154 [1],[1,3,1],[1,12,32,35,21,7,1],...; There are 35 4-covers of a labeled 3-set.
%Y A055154 Cf. A054780, A055621.
%K A055154 easy,nonn
%O A055154 1,3
%A A055154 Vladeta Jovovic (vladeta@Eunet.yu), Jun 14 2000
%I A015112
%S A015112 1,1,1,1,3,1,1,13,13,1,1,51,221,51,1,1,205,3485,3485,205,1,1,819,
%T A015112 55965,219555,55965,819,1,1,3277,894621,14107485,14107485,894621,3277,
%U A015112 1,1,13107,14317213,901984419,3625623645,901984419,14317213,13107,1,1
%V A015112 1,1,1,1,-3,1,1,13,13,1,1,-51,221,-51,1,1,205,3485,3485,205,1,1,-819,
%W A015112 55965,-219555,55965,-819,1,1,3277,894621,14107485,14107485,894621,3277,
%X A015112 1,1,-13107,14317213,-901984419,3625623645,-901984419,14317213,-13107,1,1
%N A015112 Triangle of q-binomial coefficients for q=-4.
%K A015112 sign,done,tabl,easy
%O A015112 0,5
%A A015112 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A006956 M2211
%S A006956 1,1,1,1,3,1,1,15,1,5,21,5,1,21,1,1,231,5,1,1365,1,55,21,1,1,663,11,5,57,5,1,
%T A006956 15015,1,17,483,1,11,25935,1,5,21,935,1,7917,1,23,19437,5,1,3315,1,55,21,1,1
%N A006956 Denominators of asymptotic expansion of polygamma function psi'''(z).
%D A006956 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 260, (6.4.14).
%Y A006956 Cf. A006955.
%K A006956 nonn,frac,more
%O A006956 3,5
%A A006956 sp
%E A006956 I would like to also get the sequence of numerators.
%I A060325
%S A060325 3,1,1,16,1,28,1,40,101,1,133,76,1,88,197,221,1,253,136,1,301,160,341,
%T A060325 554,196,1,208,1,220,1433,256,533,1,1147,1,613,637,328,677,701,1,1483,
%U A060325 1,388,1,2044,2164,448,1,460,941,1,1963,1013,1037,1061,1,1093,556,1
%V A060325 -3,-1,-1,16,-1,28,-1,40,101,-1,133,76,-1,88,197,221,-1,253,136,-1,301,160,341,554,
%W A060325 196,-1,208,-1,220,1433,256,533,-1,1147,-1,613,637,328,677,701,-1,1483,-1,388,-1,2044,
%X A060325 2164,448,-1,460,941,-1,1963,1013,1037,1061,-1,1093,556,-1,2299,3593,616,-1,628,3881
%N A060325 a(n) = n-th prime p(n) subtracted from sum of all composites between p(n) and p(n-1).
%e A060325 a(2) = 0 - 3 = -3. a(5) = (8+9+10) - 11 = 16.
%K A060325 nonn,easy
%O A060325 2,1
%A A060325 Jason Earls (jcearls@kskc.net), Apr 10 2001
%E A060325 Corrected and extended by Larry Reeves (larryr@acm.org), Apr 20 2001
%I A016561
%S A016561 3,1,1,21,6,1,1,21,9,2,1,1,6,1,3,102,1,1,13,1,2,1,6,1,13,
%T A016561 5,55,1,1,1,20,1,3,1,2,8,1,2,1,1,1,6,11,2,14,1,2,1,2,1,
%U A016561 2,4,4,3,1,4,2,23,1,1,11,10,3,2,2,9,17,5,2,1,7,3,16,1,2
%N A016561 Continued fraction for ln(67/2).
%K A016561 nonn,cofr
%O A016561 1,1
%A A016561 njas
%I A010272
%S A010272 3,1,1,73,14,110,2,1,18,5,4,1,9,147,8,1,10,1,4,3,5,1,1,
%T A010272 1,1,1,2,3,1,45,1,1,2,1,1,9,1,7,6,1,2,1,59,3,1,5,7,1,1,
%U A010272 1,2,2,8,2,3,9,1,8,11,1,2,5,1,1,2,249,1,2,1,5,2,3,7,1,1
%N A010272 Continued fraction for cube root of 43.
%K A010272 nonn,cofr
%O A010272 0,1
%A A010272 njas
%I A054025
%S A054025 0,1,0,1,0,0,0,3,1,2,0,4,0,0,0,1,0,3,0,0,0,0,0,4,1,2,0,2,0,0,0,3,0,2,0,
%T A054025 1,0,0,0,2,0,0,0,0,0,0,0,4,0,3,0,2,0,0,0,0,0,2,0,0,0,0,2,1,0,0,0,0,0,0,
%U A054025 0,3,0,2,4,2,0,0,0,6,1,2,0,8,0,0,0,4,0,6,0,0,0,0,0,0,0,3,0,1,0,0,0,2,0
%N A054025 Sum of divisors of n read modulo (number of divisors of n).
%F A054025 a(n) = A000203(n) mod A000005(n), sigma(n) mod tau(n)
%p A054025 with(numtheory): seq(sigma(i) mod tau(i),i=1..120);
%Y A054025 Cf. A000005, A000203.
%K A054025 nonn
%O A054025 1,8
%A A054025 Asher Auel (asher.auel@reed.edu) Jan 19, 2000
%I A054869
%S A054869 3,1,2,0,5,3,1,2,2,2,5,1,5,5,1,4,1,3,1,2,5,5,5,0,5,2,5,5,5,3,1,4,3,
%T A054869 3,0,4,2,2,4,0,1,3,3,1,4,0,2,0,1,2,5,2,4,0,2,3,3,0,3,4,5,5,2,5,5,4,
%U A054869 3,2,3,1,5,4,5,4,0,1,1,0,4,2,0,1,3,0,1,5,0,4,3,5,0,1,0,2,4,0,3,4,2
%N A054869 Digits of an idempotent 6-adic number.
%C A054869 ( a(0)+a(1)*6+a(2)*6^2+ ... )^k=a(0)+a(1)*6+a(2)*6^2+... for each k. Apart from 0 and 1, in base 6 there are only 2 numbers with this property. For the other see A055620.
%K A054869 nonn
%O A054869 0,1
%A A054869 Paolo Dominici (pd@full-service.it), May 23 2000
%I A056931
%S A056931 0,0,0,0,0,1,1,0,3,1,2,1,0,1,2,1,3,2,0,1,1,4,2,2,0,3,1,0,0,
%T A056931 2,3,0,3,0,0,0,3,0,5,4,6,5,3,0,6,1,2,6,2,2,1,2,0,1,9,0,2,2,
%U A056931 3,2,1,9,1,1,2,1,6,6,1,3,0,0,0,6,1,3,3,2,7,1,2,1,2,1,4
%V A056931 0,0,0,0,0,-1,-1,0,3,-1,-2,-1,0,1,2,1,-3,-2,0,1,1,-4,2,-2,0,3,-1,0,0,
%W A056931 -2,-3,0,-3,0,0,0,3,0,5,-4,-6,-5,-3,0,-6,1,-2,6,2,-2,1,-2,0,1,9,0,2,-2,
%X A056931 -3,2,-1,-9,1,1,2,-1,-6,-6,-1,-3,0,0,0,6,-1,-3,3,-2,-7,1,-2,1,2,-1,-4
%N A056931 Difference between n-th pronic [=n(n+1)] and average of smallest prime greater than n^2 and largest prime less than (n+1)^2.
%C A056931 a(1)=-0.5 which is not an integer
%F A056931 a(n) =A002378(n)-(A007491(n)+A053001(n+1))/2 =A002378(n)-A056930(n).
%e A056931 a(4)=0 because smallest prime greater than 4^2 is 17, largest prime less than 5^2 is 23, average of 17 and 23 is 20, and 4*5-20=0
%p A056931 with(numtheory): A056931:=n-> n*(n+1)-(prevprime((n+1)^2)+nextprime(n^2))/2);
%Y A056931 Cf. A002378, A007491, A053000, A053001, A056927, A056928, A056929, A056930.
%K A056931 easy,sign,done
%O A056931 2,9
%A A056931 Henry Bottomley (se16@btinternet.com), Jul 12 2000
%E A056931 More terms from James A. Sellers (sellersj@math.psu.edu), Jul 13 2000
%I A016569
%S A016569 3,1,2,1,1,1,4,1,1,2,11,2,5,1,3,30,1,4,1,13,2,2,2,1,1,3,
%T A016569 4,14,13,2,1,1,1,1,1,10,1,2,1,23,1,2,2,17,1,1,3,52,1,32,
%U A016569 1,1,1,3,2,1,2,1,1,1,8,6,1,2,1,6,1,15,2,2,1,6,3,7,35,1
%N A016569 Continued fraction for ln(83/2).
%K A016569 nonn,cofr
%O A016569 1,1
%A A016569 njas
%I A010281
%S A010281 3,1,2,1,2,1,4,1,2,19,7,6,3,2,3,3,1,7,1,1,1,1,13,17,1,2,
%T A010281 7,1,1,2,1,3,11,2,2,1,10,2,1,5,1,6,1,1,1,1,2,241,2,4,1,
%U A010281 6,1,2,27,1,5,2,4,2,1,1,150,12,1,1,1,2,1,2,9,6,1,2,15,1
%N A010281 Continued fraction for cube root of 52.
%K A010281 nonn,cofr
%O A010281 0,1
%A A010281 njas
%I A030777
%S A030777 3,1,2,1,3,1,3,6,2,5,1,1,7,4,6,1,3,2,1,8,5,9,1,1,2,4,4,2,10,7,
%T A030777 12,1,1,2,2,4,5,5,5,11,10,15,1,2,1,3,3,3,5,6,9,7,12,13,18,1,2,
%U A030777 1,4,2,4,5,4,7,8,11,8,16,15,21,1,2,1,4,2,5,4,5,6,7,9,9,13,12
%N A030777 Row 1, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in rows 1 and 2, for j=m,m-1,...2,1, where m=max number in row 1 from stages 1 to k-1; stage 1 is 3 in row 1.
%K A030777 nonn
%O A030777 1,1
%A A030777 Clark Kimberling, ck6@cedar.evansville.edu
%I A056595
%S A056595 0,1,1,1,1,3,1,2,1,3,1,4,1,3,3,2,1,4,1,4,3,3,1,6,1,3,2,4,1,7,1,3,3,3,3,
%T A056595 5,1,3,3,6,1,7,1,4,4,3,1,7,1,4,3,4,1,6,3,6,3,3,1,10,1,3,4,3,3,7,1,4,3,
%U A056595 7,1,8,1,3,4,4,3,7,1,7,2,3,1,10,3,3,3,6,1,10,3,4,3,3,3,9,1,4,4,5,1,7,1
%N A056595 Number of nonsquare divisors of n.
%D A056595 a(n)=A000005(n)- A046951(n)=d[n]-d[A000188(n)]
%Y A056595 A000005, A000188, A046951.
%K A056595 nonn
%O A056595 1,6
%A A056595 Labos E. (labos@ana1.sote.hu), Jul 21 2000
%I A029351
%S A029351 1,0,0,0,1,0,1,1,1,0,1,1,3,1,2,1,3,1,4,3,4,2,4,3,7,4,6,
%T A029351 4,8,4,9,7,10,6,10,8,14,9,13,10,16,10,18,14,19,13,20,16,
%U A029351 25,18,24,19,28,20,31,25,33,24,34,28,41,31,40,33,45,34
%N A029351 Expansion of 1/((1-x^4)(1-x^6)(1-x^7)(1-x^12)).
%K A029351 nonn
%O A029351 0,13
%A A029351 njas
%I A035115
%S A035115 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,2,1,3,2,
%T A035115 3,1,8,9,5,17,8,5,9,4,37,9,7,19,19,121,10,11,55,55,11,211,43,69,201,
%U A035115 695,64,351,13,468,39,507,156,84,75,4889,2593,1536,10752,41421,76301,1280,6795
%N A035115 Relative class number h- of cyclotomic field Q(zeta_m) where m is n-th term of A035113.
%D A035115 L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353.
%Y A035115 Cf. A000010, A035113, A035114, A055513, A061494, A061653.
%K A035115 nonn,nice
%O A035115 1,28
%A A035115 njas
%I A029329
%S A029329 1,0,0,0,1,1,1,0,1,1,3,1,2,1,3,3,4,2,4,3,7,4,6,4,8,7,9,
%T A029329 6,10,8,14,9,13,10,16,14,18,13,19,16,25,18,24,19,28,25,
%U A029329 31,24,33,28,40,31,40,33,45,40,49,40,52,45,61,49,61,52
%N A029329 Expansion of 1/((1-x^4)(1-x^5)(1-x^6)(1-x^10)).
%K A029329 nonn
%O A029329 0,11
%A A029329 njas
%I A046645
%S A046645 0,1,1,3,1,2,1,4,3,2,1,4,1,2,2,7,1,4,1,4,2,2,1,5,3,2,4,4,1,3,1,
%T A046645 8,2,2,2,6,1,2,2,5,1,3,1,4,4,2,1,8,3,4,2,4,1,5,2,5,2,2,1,5,1,2,
%U A046645 4,10,2,3,1,4,2,3,1,7,1,2,4,4,2,3,1,8,7,2,1,5,2,2,2,5,1,5,2,4,2
%N A046645 Log_2 A046644[ n ].
%Y A046645 See A046643 for more details.
%K A046645 nonn
%O A046645 1,4
%A A046645 njas
%I A016470
%S A016470 3,1,2,1,4,3,7,3,3,5,2,2,2,1,1,1,2,2,69,4,2,1,2,1,5,14,
%T A016470 6,1,1,1,6,1,1,1,6,10,1,9,2,4,1,23,1,2,1,19,5,37,5,20,21,
%U A016470 1,1,2,2,1,1,3,1,6,1,3,9,1,9,1,2,1,3,1,14,2,4,1,1,2,12
%N A016470 Continued fraction for ln(42).
%K A016470 nonn,cofr
%O A016470 1,1
%A A016470 njas
%I A059807
%S A059807 1,1,1,1,1,3,1,2,1,5,1
%N A059807 Maximal size of the commutator subgroup of G where G is a finite group of order n.
%C A059807 a(n) = 1 iff n belongs to sequence A051532 - Avi Peretz (njk@netvision.net.il), Feb 27 2001
%e A059807 a(6) = 3 because the commutator subgroup of the symmetric group S_3 is the group Z_3.
%Y A059807 Cf. A059806, A051532.
%K A059807 nonn
%O A059807 0,6
%A A059807 Noam Katz (noamkj@hotmail.com), Feb 24 2001
%I A010123
%S A010123 3,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,
%T A010123 1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,
%U A010123 2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6
%N A010123 Continued fraction for sqrt(14).
%H A010123 Index entries for continued fractions for constants
%K A010123 nonn
%O A010123 0,1
%A A010123 njas
%I A039620
%S A039620 1,1,1,1,3,1,2,1,6,1,6,0,4,5,10,1,24,4,15,25,15,1,120,28,49,35,
%T A039620 70,21,1,720,188,196,49,0,154,28,1,5040,1368,944,0,231,252,294,36,
%U A039620 1,40320,11016,5340,820,1365,987,1050,510,45,1
%V A039620 1,1,1,-1,3,1,2,-1,6,1,-6,0,4,5,10,1,24,4,-15,25,15,1,-120,-28,49,-35,
%W A039620 70,21,1,720,188,-196,49,0,154,28,1,-5040,-1368,944,0,-231,252,294,36,
%X A039620 1,40320,11016,-5340,-820,1365,-987,1050,510,45,1
%N A039620 Triangle of Lehmer-Comtet numbers of 1st kind.
%C A039620 D.H.Lehmer, "Numbers Associated with Stirling Numbers and x^x", Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
%D A039620 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139
%F A039620 E.g.f. for a(n,k): (1/k!)[ (1+x)*ln(1+x) ]^k
%e A039620 1; 1,1; -1,3,1; 2,-1,6,1; -6,0,5,10,1; etc.
%Y A039620 Cf. A039621.
%K A039620 nonn,tabl
%O A039620 1,5
%A A039620 Leonard Smiley (smiley@math.uaa.alaska.edu)
%I A008296
%S A008296 1,1,1,1,3,1,2,1,6,1,6,0,5,10,1,24,4,15,25,15,1,120,28,49,35,70,21,1,
%T A008296 720,188,196,49,0,154,28,1,5040,1368,944,0,231,252,294,36,1,40320,
%U A008296 11016,5340,820,1365,987,1050,510,45,1,362880,98208,34716,9020,7645
%V A008296 1,1,1,-1,3,1,2,-1,6,1,-6,0,5,10,1,24,4,-15,25,15,1,-120,-28,49,-35,70,21,1,720,188,
%W A008296 -196,49,0,154,28,1,-5040,-1368,944,0,-231,252,294,36,1,40320,11016,-5340,-820,1365,
%X A008296 -987,1050,510,45,1,-362880,-98208,34716,9020,-7645,3003,-1617,2970,825,55,1,3628800
%N A008296 Triangle arising in expansion of ((1+x)log(1+x))^n.
%D A008296 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.
%F A008296 Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
%F A008296 a(n+1,k) = n*a(n-1,k-1) + a(n,k-1) + (k-n)*a(n,k).
%F A008296 a(n,k) = Sum_{l} binomial(l,k)*k^(l-k)*stirling1(n,l).
%e A008296 1; 1,1; -1,3,1; 2,-1,6,1; ...
%e A008296 with(combinat): for n from 1 to 20 do for k from 1 to n do printf(`%d,`,sum(binomial(l,k)*k^(l-k)*stirling1(n,l), l=k..n)) od: od:
%Y A008296 Diagonals give A000142, A045406, A000217, A059302. Row sums give A005727.
%K A008296 sign,done,tabl,easy,nice
%O A008296 1,5
%A A008296 njas
%E A008296 More terms from James A. Sellers (sellersj@math.psu.edu), Jan 26 2001
%I A011086
%S A011086 3,1,2,1,9,8,5,6,4,1,3,5,2,1,4,4,9,6,1,3,4,1,0,3,2,4,9,8,0,5,7,8,1,
%T A011086 5,8,6,3,3,9,4,5,6,1,4,0,5,3,0,1,0,6,5,9,8,0,1,4,1,3,4,8,3,0,4,5,9,
%U A011086 6,6,6,9,5,8,6,9,9,4,5,1,5,8,0,3,6,8,0,8,6,3,2,8,3,7,9,8,5,3,9,1,9
%N A011086 Decimal expansion of 4th root of 95.
%K A011086 nonn,cons
%O A011086 1,1
%A A011086 njas
%I A016570
%S A016570 3,1,2,1,125,3,1,1,1,1,6,1,2,1,2,1,2,1,5,20,1,1,1,2,3,6,
%T A016570 2,1,8,19,1,62,1,1,6,2,3,6,2,6,1,1,1,1,1,5,6,7,8,3,2,8,
%U A016570 8,1,6,3,4,1,11,1,1,1,1,3,7,4,1,2,1,4,1,2,7,1,14,22,76
%N A016570 Continued fraction for ln(85/2).
%K A016570 nonn,cofr
%O A016570 1,1
%A A016570 njas
%I A046804
%S A046804 0,0,0,0,0,1,3,1,2,2,0,2,0,1,5,2,5,0,4,0,1,7,2,8,6,0,1,2,1,2,1,0,4,4,5,
%T A046804 0,3,1,6,2,8,0,0,1,1,1,0,1,3,4,2,5,0,0,5,2,8,0,4,0,1,2,6,0,1,2,0,1,4,7,
%U A046804 2,8,3,1,1,2,2,5,0,4,5,0,0,1,7,2,8,2,0,1,5,2,4,0,4,2,5,0,1,0,1,4,2,2,0
%N A046804 Primes p modulo t where t = terminal digit of p.
%D A046804 Idea derived from "The Creation of New Mathematics: An Application of the Lakatos Heuristic," pp 292-298 of Philip J. Davis and Reuben Hersh, The Mathematical Experience, Houghton Mifflin Co, 1982. ISBN 0-395-32131-X.
%H A046804 Index entries for sequences related to final digits of numbers
%e A046804 If p = 29, then 29 - 27 = 2.
%K A046804 easy,nonn
%O A046804 0,7
%A A046804 Enoch Haga (EnochHaga@msn.com)
%I A056529
%S A056529 0,3,1,2,2,1,1,2,0,1,3,1,0,1,1,0,3,1,2,3,1,1,1,2,1,3,2,3,1,1,1,2,2,1,2,
%T A056529 2,3,1,1,2,1,2,2,3,1,2,1,1,2,2,1,1,3,2,2,1,2,1,1,1,1,3,3,3,1,2,2,1,2,1,
%U A056529 2,2,3,2,2,2,2,2,1,2,2,3,2,2,1,2,3,2,3,1,3,2,3,2,1,2,2,3,2,1,3,1,1,1,1
%N A056529 Number of iterations of sum of digits of square to reach 1, 9, 13 or 16.
%e A056529 a(2)=3 because successive iterations give 2 -> 4 -> 7 -> 13 -> 16 -> 13 etc.
%Y A056529 Cf. A004159 for sum of digits of square, A056528 for two iterations, A056020 where iteration settles to 1, A056020 where iteration settles to 9, A056527 where iteration settles to 13 and 16.
%K A056529 base,easy,nonn
%O A056529 1,2
%A A056529 Henry Bottomley (se16@btinternet.com), Jun 19 2000
%I A016568
%S A016568 3,1,2,2,1,6,1,50,1,1,1,45,1,3,4,1,1,5,27,1,10,1,1,1,1,
%T A016568 18,1,27,7,1,1,1,3,1,49,2,9,1,20,1,5,2,1,1,2,7,3,1,36,1,
%U A016568 2,6,1,3,26,1,54,18,1,9,4,3,1,1,16,1,1,4,1,5,4,2,2,2,3
%N A016568 Continued fraction for ln(81/2).
%K A016568 nonn,cofr
%O A016568 1,1
%A A016568 njas
%I A021888
%S A021888 0,0,1,1,3,1,2,2,1,7,1,9,4,5,7,0,1,3,5,7,4,6,6,0,6,3,3,4,8,4,1,6,2,
%T A021888 8,9,5,9,2,7,6,0,1,8,0,9,9,5,4,7,5,1,1,3,1,2,2,1,7,1,9,4,5,7,0,1,3,
%U A021888 5,7,4,6,6,0,6,3,3,4,8,4,1,6,2,8,9,5,9,2,7,6,0,1,8,0,9,9,5,4,7,5,1
%N A021888 Decimal expansion of 1/884.
%K A021888 nonn,cons
%O A021888 0,5
%A A021888 njas
%I A038575
%S A038575 1,1,1,3,1,2,2,2,1,6,1,2,3,3,1,5,2,4,3,2,1,9,3,2,4,4,1,7,2,4,3,2,3,10,
%T A038575 3,3,3,6,2,7,1,5,5,3,1,12,3,6,3,4,2,8,4,7,5,3,2,12,2,3,5,6,3,7,3,5,5,7,
%U A038575 2,14,2,4,6,5,4,8,2,9,7,3,1,13,4,3,4,9,2,12,5,6,4,2,6,16,4,5,6,10,2,8
%N A038575 Number of prime factors of n-th Fibonacci number, with multiplicity.
%K A038575 nonn
%O A038575 3,4
%A A038575 Jeff Burch (gburch@erols.com)
%I A033178
%S A033178 1,1,1,3,1,2,2,2,2,3,2,4,2,2,2,4,2,4,2,4,2,4,1,5,4,3,3,5,2,4,3,5,2,3,2,
%T A033178 6,3,3,4,7,2,5,2,4,4,5,2,5,4,4,3,7,2,5,4,5,4,4,2,9,3,4,4,7,2,5,5,4,3,6,
%U A033178 3,9,4,3,3,6,3,5,2,7,4,5,2,10,5,4,5,8,2,6,3,6,3,6,5,6,5,4,5,8,3,6,3,5
%N A033178 Sorted sequences of positive integers of length n having equal sum and product.
%D A033178 Comment from Mario Velucchi (velucchi@CLI.DI.Unipi.IT): See 'Unsol.Prob in Number Theory' 2 ed by R. Guy (Section D24) and the investigation by Prof. David Singmaster ...
%Y A033178 Cf. A033179.
%K A033178 nonn
%O A033178 2,4
%A A033178 dww
%I A029418
%S A029418 1,0,0,0,0,0,1,1,0,1,0,1,1,1,1,1,1,1,3,1,2,2,2,2,3,3,2,
%T A029418 4,3,4,4,4,4,5,5,5,7,5,6,7,7,7,9,8,8,10,9,10,11,11,11,13,
%U A029418 12,13,15,14,15,16,16,16,19,18,19,21,20,21,23,23,23,26
%N A029418 Expansion of 1/((1-x^6)(1-x^7)(1-x^9)(1-x^11)).
%K A029418 nonn
%O A029418 0,19
%A A029418 njas
%I A029336
%S A029336 1,0,0,0,1,1,0,1,1,1,1,1,3,1,2,2,3,3,2,4,4,4,4,4,7,5,6,
%T A029336 6,8,8,7,9,10,10,10,11,14,12,13,14,17,16,16,18,20,20,20,
%U A029336 22,25,24,25,26,30,29,30,32,35,35,35,38,42,41,42,44,49
%N A029336 Expansion of 1/((1-x^4)(1-x^5)(1-x^7)(1-x^12)).
%K A029336 nonn
%O A029336 0,13
%A A029336 njas
%I A010280
%S A010280 3,1,2,2,3,17,1,2,2,5,1,12,1,1,5,1,1,1,6,2,1,1,1,2,1,5,
%T A010280 4,2,5,3,2,19,8,7038,19,1,3,2,1,2,2,4,2,7,5,1,5,1,2,2,9,
%U A010280 1,1,5,1,1,1,13,11,3,1,1,1,1,10,6,1,3,2,1,4,1,8,3,2,1,1
%N A010280 Continued fraction for cube root of 51.
%K A010280 nonn,cofr
%O A010280 0,1
%A A010280 njas
%I A002016 M2212 N0878
%S A002016 1,3,1,2,2,4,2,6,1,8,2,10,2,5,4,14,3,16,2,7,4,20,4,10,5,18,4,26,2,
%T A002016 28,8,16,7,8,6,34,8,20,4,38,3,40,8,12,10,44,8,28,5,30,10,50,9,16,8,33,13
%N A002016 Number of first n tetrahedral numbers prime to n.
%D A002016 Problem 272, Amer. Math. Monthly, 41 (1934), 582-587.
%K A002016 nonn
%O A002016 1,2
%A A002016 njas
%I A057056
%S A057056 1,3,1,2,2,11,8,1,7,12,15,15,11,2,24,5,22,37,49,1,60,57,47,29,2,46,2,
%T A057056 36,66,91,5,12,11,1,106,80,42,132,72,149,63,123,7,46,76,96,105,102,86,
%U A057056 56,11,173,101,11,144,21,132,234,57,131
%N A057056 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6;...; each k is an R(i(k),j(k)), and A057056(n)=j(C(n,3)).
%K A057056 nonn
%O A057056 3,2
%A A057056 Clark Kimberling, ck6@cedar.evansville.edu, Jul 30 2000
%I A016469
%S A016469 3,1,2,2,28,5,1,15,6,1,25,1,54,1,5,10,2,2,4,1,4,2,1,1,1,
%T A016469 3,2,1,1,1,1,5,5,1,2,22,3,36,2,5,1,3,1,3,1,5,1,8,2,1,1,
%U A016469 3,1,1,1,6,2,1,8,1,2,9,1,68,1,2,1,1,10,2,39,3,2,13,14,3
%N A016469 Continued fraction for ln(41).
%K A016469 nonn,cofr
%O A016469 1,1
%A A016469 njas
%I A004591
%S A004591 3,1,2,3,0,5,4,0,7,2,6,6,4,5,5,5,2,2,4,4,4,0,2,2,4,2,5,7,1,0,1,4,1,
%T A004591 4,6,6,3,3,7,7,5,2,2,5,3,2,3,4,0,5,2,7,2,7,6,0,6,0,0,0,1,6,1,2,7,2,
%U A004591 5
%N A004591 Expansion of sqrt(10) in base 8.
%K A004591 nonn,base,cons
%O A004591 1,1
%A A004591 njas
%I A036584
%S A036584 3,1,2,3,2,1,3,1,2,1,3,2,3,1,2,3,2,1,3,2,3,1,2,1,3,1,2,3,2,1,3,1,2,
%T A036584 1,3,2,3,1,2,1,3,1,2,3,2,1,3,2,3,1,2,3,2,1,3,1,2,1,3,2,3,1,2,3,2,1,
%U A036584 3,2,3,1,2,1,3,1,2,3,2,1,3,2,3,1,2,3,2,1,3,1,2,1,3,2,3,1,2,1,3,1,2
%N A036584 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.
%D A036584 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.
%K A036584 nonn
%O A036584 0,1
%A A036584 njas
%I A029279
%S A029279 1,0,0,1,0,1,1,1,1,1,3,1,2,3,2,4,3,4,4,4,7,5,6,7,7,9,8,
%T A029279 10,10,10,14,12,13,15,15,18,17,19,20,20,25,23,25,27,27,
%U A029279 32,30,33,35,35,41,39,42,44,45,51,49,53,55,56,63,61,65
%N A029279 Expansion of 1/((1-x^3)(1-x^5)(1-x^7)(1-x^10)).
%K A029279 nonn
%O A029279 0,11
%A A029279 njas
%I A026181
%S A026181 3,1,2,3,3,1,2,1,2,3,1,2,3,3,1,2,3,3,1,2,1,2,3,1,2,1,2,3,1,2,
%T A026181 3,3,1,2,1,2,3,1,2,3,3,1,2,3,3,1,2,1,2,3,1,2,3,3,1,2,3,3,1,2,
%U A026181 1,2,3,1,2,1,2,3,1,2,3,3,1,2,1,2,3,1,2,1,2,3,1,2,3,3,1,2,1,2
%N A026181 a(n) = (1/2)*(s(n) - s(n-1)), where s = A026180.
%K A026181 nonn
%O A026181 3,1
%A A026181 Clark Kimberling, ck6@cedar.evansville.edu
%I A021766
%S A021766 0,0,1,3,1,2,3,3,5,9,5,8,0,0,5,2,4,9,3,4,3,8,3,2,0,2,0,9,9,7,3,7,5,
%T A021766 3,2,8,0,8,3,9,8,9,5,0,1,3,1,2,3,3,5,9,5,8,0,0,5,2,4,9,3,4,3,8,3,2,
%U A021766 0,2,0,9,9,7,3,7,5,3,2,8,0,8,3,9,8,9,5,0,1,3,1,2,3,3,5,9,5,8,0,0,5
%N A021766 Decimal expansion of 1/762.
%K A021766 nonn,cons
%O A021766 0,4
%A A021766 njas
%I A050056
%S A050056 1,3,1,2,3,4,5,8,13,14,15,18,23,36,51,74,125,126,127,130,135,148,163,
%T A050056 186,237,362,489,624,787,1024,1513,2300,3813,3814,3815,3818,3823,3836,
%U A050056 3851,3874,3925,4050,4177,4312,4475,4712,5201
%N A050056 a(n)=a(n-1)+a(m), where m=2n-3-2^(p+1), and 2^p= 4.
%K A050056 nonn
%O A050056 1,2
%A A050056 Clark Kimberling, ck6@cedar.evansville.edu
%I A060477
%S A060477 3,1,2,3,6,9,18,30,56,99,186,335,630,1161,2182,4080,7710,14532,27594,
%T A060477 52377,99858,190557,364722,698870,1342176,2580795,4971008,9586395,
%U A060477 18512790,35790267,69273666,134215680,260300986
%N A060477 Number of orbits of length n in map whose periodic points are A048578.
%D A060477 Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
%H A060477 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%F A060477 If b(n) is the n-th term of A048578, then a(n)=(1/n)*\sum_{d|n}\mu(d)a(n/d)
%e A060477 a(3)=2 since the 3rd term of A048578 is 9, and the 1st term is 3.
%Y A060477 A048578.
%K A060477 easy,nonn
%O A060477 1,1
%A A060477 Thomas Ward (t.ward@uea.ac.uk)
%I A016468
%S A016468 3,1,2,4,1,2,51,1,68,1,1,16,1,1,1,6,21,4,1,1,3,1,1,1,9,
%T A016468 1,16,2,8,3,2,9,11,1,1,3,1,1,1,1,4,1,4,6,4,1,2,1,3,1,1,
%U A016468 20,8,1,2,75,1,1,1,3,1,2,1,14,1,1,1,2,5,6,1,1,1,13,2,3
%N A016468 Continued fraction for ln(40).
%K A016468 nonn,cofr
%O A016468 1,1
%A A016468 njas
%I A059016
%S A059016 1,0,0,1,0,1,3,1,2,4,1,3,6,3,3,6,2,4,8,8,5,8,7,4,10,11,8,7,8,7,12,10,
%T A059016 13,9,11,13,12,11,16,14,11,11,14,13,12,16,10,19,21,15,16,18,18,19,21,
%U A059016 16,17,23,16,20,25,23,16,20,24,19,26,20,32,24,25,27,24,23,27,28,29,31
%N A059016 Number of 0's in binary expansion of Fibonacci(n).
%Y A059016 Cf. A011373.
%K A059016 nonn,base
%O A059016 0,7
%A A059016 Patrick De Geest (pdg@worldofnumbers.com), Jan 2001. II.
%I A004608
%S A004608 3,1,2,4,1,8,8,1,2,4,0,7,4,4,2,7,8,8,6,4,5,1,7,7,7,6,1,7,3,1,0,3,5,
%T A004608 8,2,8,5,1,6,5,4,5,3,5,3,4,6,2,6,5,2,3,0,1,1,2,6,3,2,1,4,5,0,2,1,4,
%U A004608 1
%N A004608 Expansion of Pi in base 9.
%K A004608 nonn,base,cons
%O A004608 1,1
%A A004608 njas
%I A049992
%S A049992 0,0,1,1,1,3,1,2,4,3,1,7,1,3,8,4,1,10,1,6,10,4,1,14,4,4,12,7,1,19,1,6,
%T A049992 14,5,7,22,1,5,16,12,1,24,1,8,25,6,1,27,4,12,21,9,1,29,9,12,23,7,1,40,
%U A049992 1,7,30,11,10,35,1,10,27,21,1,42,1,8,39,11
%N A049992 a(n)=number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum n.
%K A049992 nonn
%O A049992 1,6
%A A049992 Clark Kimberling, ck6@cedar.evansville.edu
%I A021036
%S A021036 0,3,1,2,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A021036 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A021036 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A021036 Decimal expansion of 1/32.
%K A021036 nonn,cons
%O A021036 0,2
%A A021036 njas
%I A033765
%S A033765 1,1,1,3,1,2,5,2,3,7,4,4,10,3,3,11,6,4,12,6,5,19,6,8,16,
%T A033765 7,10,17,7,8,25,10,9,20,8,8,27,12,11,30,11,14,27,12,14,
%U A033765 29,14,12,37,15,11
%N A033765 Product t2(q^d); d | 6, where t2 = theta2(q)/(2*q^(1/4)).
%K A033765 nonn
%O A033765 0,4
%A A033765 njas
%I A033777
%S A033777 1,1,1,3,1,2,5,2,3,8,5,5,13,4,5,16,8,7,20,11,10,32,10,13,
%T A033777 32,15,17,38,19,19,60,21,24,57,25,28,70,33,32,94,33,37,
%U A033777 85,39,40,98,46,42,133,49,52
%N A033777 Product t2(q^d); d | 18, where t2 = theta2(q)/(2*q^(1/4)).
%K A033777 nonn
%O A033777 0,4
%A A033777 njas
%I A033801
%S A033801 1,1,1,3,1,2,5,3,4,8,7,5,12,8,6,15,14,11,17,18,13,27,23,
%T A033801 24,33,26,32,40,36,35,61,45,41,68,44,47,79,72,59,88,73,
%U A033801 67,111,87,100,125,99,109,146,108,119
%N A033801 Product t2(q^d); d | 42, where t2 = theta2(q)/(2*q^(1/4)).
%K A033801 nonn
%O A033801 0,4
%A A033801 njas
%I A050058
%S A050058 1,3,1,2,5,6,9,10,12,13,16,17,19,24,30,39,49,50,53,54,56,61,67,76,86,
%T A050058 98,111,127,144,163,187,217,256,257,260,261,263,268,274,283,293,305,
%U A050058 318,334,351,370,394,424
%N A050058 a(n)=a(n-1)+a(m), where m=n-1-2^p, and 2^p= 4.
%K A050058 nonn
%O A050058 1,2
%A A050058 Clark Kimberling, ck6@cedar.evansville.edu
%I A048226
%S A048226 3,1,2,5,9,10,12,13,15,18,21,23,29,32,34,37,41,43,45,47,48,53,56,57,60,
%T A048226 63,65,69,72,74,75,76,80,81,83,86,87,93,94,96,97,99,100,104,107,108,
%U A048226 110,111,114,115,116,120,122,124,125,127,130
%N A048226 a(n)=T(n,1), array T given by A048225.
%K A048226 nonn
%O A048226 1,1
%A A048226 Clark Kimberling, ck6@cedar.evansville.edu
%I A049919
%S A049919 1,3,1,2,5,9,19,31,40,108,217,427,832,1587,2855,4550,6137,16821,33643,
%T A049919 67279,134536,268995,537671,1074182,2145401,4278531,8506604,16811492,
%U A049919 32817797,62431245,112329529,179039305,241470550,661980402
%N A049919 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=2n-2-2^(p+1), and 2^pLink to a section of The World of Mathematics.
%Y A052417 Cf. A052415, ..., A052426.
%K A052417 nonn,hard
%O A052417 1,9
%A A052417 Eric W. Weisstein (eric@weisstein.com)
%I A055179
%S A055179 3,1,2,6,5,7,4,8,9,10,12,11,15,18,13,16,21,17,24,27,19,30,20,32,22,34,
%T A055179 23,37,26,14,25,41,35,28,29,44,40,31,50,42,33,54,48,38,58,52,45,62,55,
%U A055179 65,59,49,51,68,63,36,43,53,72,66,39,78,71
%N A055179 n-th distinct number to appear in A055171; also the n-th to appear in A055191.
%C A055179 Conjecture: this sequence is a permutation of the positive integers.
%K A055179 nonn
%O A055179 1,1
%A A055179 Clark Kimberling, ck6@cedar.evansville.edu, Apr 27 2000
%I A049917
%S A049917 1,3,1,2,6,11,23,44,90,137,295,602,1209,2422,4845,9688,19378,29069,
%T A049917 62981,128385,257983,516573,1033453,2067064,4134175,8268396,16536813,
%U A049917 33073638,66147281,132294566,264589133,529178264,1058356530
%N A049917 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=2^(p+1)+2-n, and 2^pEncyclopedia of Combinatorial Structures 896
%F A052914 G.f.: -(-1+x)/(1-x-2*x^4+2*x^5-x^3)
%F A052914 Recurrence: {a(1)=0,a(0)=1,a(2)=0,a(3)=1,a(4)=3,2*a(n)-2*a(n+1)-a(n+2)-a(n+4)+a(n+5)}
%F A052914 Sum(-1/19913*(-418-4709*_alpha+599*_alpha^2+1048*_alpha^3+542*_alpha^4)*_alpha^(-1-n),_alpha=RootOf(1-_Z-2*_Z^4+2*_Z^5-_Z^3))
%p A052914 spec:= [S,{S=Sequence(Prod(Union(Sequence(Z),Z,Z),Z,Z,Z))},unlabelled]: seq(combstruct[count](spec,size=n),n=0..20);
%K A052914 easy,nonn
%O A052914 0,5
%A A052914 encyclopedia@pommard.inria.fr, Jan 25 2000
%E A052914 More terms from James A. Sellers (sellersj@math.psu.edu), May 06 2000
%I A060750
%S A060750 3,1,2,8,5,4,78,37,6,11,28,12,349,13,383,10,18,16,29,17,33,210,14,
%T A060750 133,32,60,19,106,57,20,48,26,21,35,97,217,25,22,13932,863,205,54,
%U A060750 30452,306,2591,40,44,39,49,38,51,47,30,252992198,2253,101,112,246,402,119,53,139
%N A060750 Step at which card n appears on top of deck for first time in Guy's shuffling problem A035485.
%C A060750 Card #1 is initially at the top of the deck, and next appears at the top of the deck after 3 shuffles. Here we do not accept 0 as a valid number of shuffles, and so we say that card #1 first shows up on top after 3 shuffles. A060751 and A060752 also adopt this convention. Alternatively, we can say that card #1 first shows up on top after 0 shuffles; this leads to sequences A035490, A057983, A057984, etc.
%D A060750 See A035490 for references, links and programs.
%Y A060750 Cf. A035485, A035490-A035494, A060751, A060752.
%K A060750 nonn,nice
%O A060750 1,1
%A A060750 David Wilson, Apr 22, 2001.
%I A058142
%S A058142 1,1,1,1,3,1,2,9,6,2,1,26,30,16,5,1,98,142,111,54,15,1,455,718,713,482,
%T A058142 215,53,3
%N A058142 Triangle: Commutative monoids of order n with k idempotents.
%H A058142 Index entries for sequences related to monoids
%e A058142 1; 1,1; 1,3,1; 2,9,6,2; 1,26,30,16,5; ...
%Y A058142 Row sums give A058131. Main diagonal: A006966. Columns 1, 2: A000688, A058143.
%K A058142 nonn,tabl,more
%O A058142 1,5
%A A058142 Christian G. Bower (bowerc@usa.net), Nov 14 2000
%I A058144
%S A058144 1,1,1,1,3,1,2,9,6,2,1,28,31,18,6,2,131,160,132,65,19,1,1081,947,917,
%T A058144 612,275,68,5
%N A058144 Triangle: Self-converse monoids of order n with k idempotents.
%H A058144 Index entries for sequences related to monoids
%e A058144 1; 1,1; 1,3,1; 2,9,6,2; 1,28,31,18,6; ...
%Y A058144 Row sums give A058132. Main diagonal: A058122(n-1). Columns 1-3: A000001, A058145, A058146.
%K A058144 nonn,tabl,more
%O A058144 1,5
%A A058144 Christian G. Bower (bowerc@usa.net), Nov 14 2000
%I A055450
%S A055450 1,1,3,1,2,10,1,3,7,36,1,4,5,26,137,1,5,9,19,101,543,1,6,14,14,75,406,
%T A055450 2219,1,7,20,28,56,305,1676,9285,1,8,27,48,42,230,1270,7066,39587,1,9,
%U A055450 35,75,90,174,965,5390,30302,171369,1,10,44,110
%N A055450 Path-counting array T; each step of a path is (1 right) or (1 up) to a point below line y=x, else (1 right and 1 up) or (1 up) to a point on the line y=x, else (1 left) or (1 up) to a point above line y=x. T(i,j)=number of paths to point (i-j,j), for 1<=j<=i, i >= 1.
%F A055450 Initial values: T(i,0)=1 for i >= 0. Recurrence: if 1<=j= 0: Catalan numbers. T(n, n)=A002212(n), T(n, n-1)=A045868(n).
%Y A055450 Partial diagonals: {1}; {1, 2}; {1, 3, 5}; {1, 4, 9, 14}; ... form A030237.
%K A055450 nonn,tabl
%O A055450 0,3
%A A055450 Clark Kimberling, ck6@cedar.evansville.edu, May 18 2000
%I A016567
%S A016567 3,1,2,11,5,355,1,2,1,4,2,1,1,1,2,1,2,1,1,2,8,1,2,3,2,1,
%T A016567 3,5,4,1,11,2,3,3,1,4,2,2,1,1,15,28,5,1,2,8,2,1,6,2,6,1,
%U A016567 13,1,1,5,8,1,2,1,5,18,1,58,2,62,1,16,5,1,1,1,1,2,5,1,12
%N A016567 Continued fraction for ln(79/2).
%K A016567 nonn,cofr
%O A016567 1,1
%A A016567 njas
%I A004468
%S A004468 0,3,1,2,12,15,13,14,4,7,5,6,8,11,9,10,48,51,49,50,60,63,61,62,52,55,53,
%T A004468 54,56,59,57,58,16,19,17,18,28,31,29,30,20,23,21,22,24,27,25,26,32,35,
%U A004468 33,34,44,47,45,46,36,39,37,38,40,43,41,42,192,195,193,194,204,207,205
%N A004468 Nim product 3 * n.
%D A004468 J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%H A004468 Index entries for sequences related to Nim-multiplication
%K A004468 nonn
%O A004468 0,2
%A A004468 njas
%E A004468 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A011087
%S A011087 3,1,3,0,1,6,9,1,6,0,1,4,6,5,7,4,6,3,3,1,6,8,9,7,0,9,9,8,3,1,7,3,7,
%T A011087 9,6,1,9,6,2,1,4,5,3,8,8,5,5,4,9,5,7,8,4,4,5,6,3,9,6,1,8,1,8,4,0,2,
%U A011087 1,6,4,6,8,8,2,1,0,5,7,6,4,0,1,7,4,8,4,0,7,0,5,4,3,8,2,0,9,1,7,8,1
%N A011087 Decimal expansion of 4th root of 96.
%K A011087 nonn,cons
%O A011087 1,1
%A A011087 njas
%I A011430
%S A011430 1,3,1,3,0,3,2,4,3,4,4,8,7,8,8,3,9,2,6,4,3,0,9,3,0,5,1,0,4,3,8,0,4,
%T A011430 0,8,4,0,4,7,6,0,1,0,8,3,1,2,8,3,5,3,0,0,4,7,6,0,8,5,7,1,6,3,3,7,3,
%U A011430 7,6,4,5,1,7,5,7,8,7,7,1,7,8,3,3,0,5,7,8,4,5,5,5,0,4,6,8,9,2,6,7,6
%N A011430 Decimal expansion of 11-th root of 20.
%K A011430 nonn,cons
%O A011430 1,2
%A A011430 njas
%I A020815
%S A020815 1,3,1,3,0,6,4,3,2,8,5,9,7,2,2,5,5,6,6,4,9,3,3,4,6,7,2,8,8,2,0,4,0,
%T A020815 0,5,1,8,1,6,5,6,3,8,8,9,5,2,6,6,6,5,4,9,0,8,4,3,9,7,3,8,6,6,2,1,5,
%U A020815 8,6,8,1,4,0,6,4,4,3,6,2,0,9,0,4,2,4,8,6,6,7,1,9,2,1,3,5,7,1,5,8,0
%N A020815 Decimal expansion of 1/sqrt(58).
%K A020815 nonn,cons
%O A020815 0,2
%A A020815 njas
%I A063065
%S A063065 1,1,1,1,1,1,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,
%T A063065 1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1,1,0,1,3,1,3,1,1,1,1,1,3,1,3,1,1,1,1,
%U A063065 1,3,1,3,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1,1,2,1,2,1,2,3,1,2,3,1,1,1,0
%N A063065 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063064 in order to obtain a term in the trajectory of 10563.
%e A063065 12543 is a term of A063064. One 'Reverse and Add!' operation applied to 12543 leads to a term (47064) in the trajectory of 10563, so the corresponding term of the present sequence is 1.
%Y A063065 A023108, A033865, A063063, A063064.
%K A063065 base,nonn
%O A063065 0,12
%A A063065 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jul 07 2001
%I A051718
%S A051718 1,1,3,1,3,1,1,1,1,5,1017,691,601,7,809,3617,922191,43867,
%T A051718 6132631,174611,12988703,854513,1552922421,236364091,1139644561,
%U A051718 8553103,7089687053,23749461029,378639019356093,8615841276005
%V A051718 1,1,3,1,-3,-1,1,1,1,-5,-1017,691,601,-7,-809,3617,922191,-43867,
%W A051718 -6132631,174611,12988703,-854513,-1552922421,236364091,1139644561,
%X A051718 -8553103,-7089687053,23749461029,378639019356093,-8615841276005
%N A051718 Numerators of column 2 of table described in A051714/A051715.
%H A051718 M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
%Y A051718 Cf. A051719.
%K A051718 sign,done,easy,nice
%O A051718 0,3
%A A051718 njas
%E A051718 More terms from James A. Sellers (sellersj@math.psu.edu), Dec 08 1999
%I A016472
%S A016472 3,1,3,1,1,1,2,1,2,2,1,2,6,1,1,1,1,1,2,2,9,1,1,127,2,1,
%T A016472 1,8,1,1,2,2,1,3,1,10,1,5,2,3,2,8,1,1,19,1,3,1,2,3,1,3,
%U A016472 1,3,1,2,1,1,1,4,1,1,12,2,4,2,22,1,2,3,4,2,12,4,1,2,2,2
%N A016472 Continued fraction for ln(44).
%K A016472 nonn,cofr
%O A016472 1,1
%A A016472 njas
%I A060839
%S A060839 1,1,1,1,1,1,3,1,3,1,1,1,3,3,1,1,1,3,3,1,3,1,1,1,1,3,3,3,1,1,3,1,1,1,3,
%T A060839 3,3,3,3,1,1,3,3,1,3,1,1,1,3,1,1,3,1,3,1,3,3,1,1,1,3,3,9,1,3,1,3,1,1,3,
%U A060839 1,3,3,3,1,3,3,3,3,1,3,1,1,3,1,3,1,1,1,3,9,1,3,1,3,1,3,3,3,1,1,1,3,3,3
%N A060839 Number of non-congruent solutions of x^3 == 1 mod n .
%F A060839 Let b(n) be the number of primes dividing n which are congruent to 1 mod 3 (sequence A005088); then a(n) is 3^b(n) if n is not divisible by 9, and 3^(b(n) + 1) if n is divisible by 9.
%e A060839 a(7) = 3 because the three solutions to x^3 == 1 mod 7 are: x = 1,2,4
%Y A060839 A005088, A060594.
%K A060839 nonn,nice,easy
%O A060839 1,7
%A A060839 Ahmed Fares (ahmedfares@my-deja.com), May 02 2001
%E A060839 More terms from Larry Reeves (larryr@acm.org), May 03 2001
%I A010283
%S A010283 3,1,3,1,1,5,1,1,1,27,1,4,3,3,2,4,1,2,1,3,2,11,11,1,1,1,
%T A010283 7,4,1,3,2,3,1,47,1,3,1,2,1,177,1,1,17,1,1,40,4,4,2,2,3,
%U A010283 10,4,1,13,4,1,2,2,2,7,2,2,1,2,1,3,1,21,2,5,7,3,10,1,1
%N A010283 Continued fraction for cube root of 54.
%K A010283 nonn,cofr
%O A010283 0,1
%A A010283 njas
%I A003636 M2213
%S A003636 1,1,1,1,1,1,1,1,3,1,3,1,2,1,1,5,1,1,2,2,3,1,2,7,5,3,1,3,1,1,4,5,3,3,4,
%T A003636 1,3,5,1,1,5,5,1,2,3,5,1,7,3,2,5,1,4,11,1,5,4,2,1,13,1,9,2,3,3,7,2,7,5,
%U A003636 1,3,1,1,15,3,3,4,7,3,2,2,13,2,1,11,2,1,3,7,2,2,4,5,4,5,3,2,19,1,5,2,6
%N A003636 Classes per genus in quadratic field with discriminant -n.
%D A003636 D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
%K A003636 nonn
%O A003636 3,9
%A A003636 njas,mb
%I A030728
%S A030728 3,1,3,1,2,3,1,2,3,1,2,3,5,6,1,2,3,4,5,6,7,1,2,3,4,5,6,7,8,9,
%T A030728 1,2,3,4,5,6,7,8,9,10,12,1,2,3,4,5,6,7,8,9,10,11,12,15,1,2,3,
%U A030728 4,5,6,7,8,9,10,11,12,13,15,18,1,2,3,4,5,6,7,8,9,10,11,12,13
%N A030728 Row 2, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in rows 1 and 2, for j=1,2,...,m, where m=max number in row 1 from stages 1 to k-1; stage 1 is 3 in row 1.
%K A030728 nonn
%O A030728 1,1
%A A030728 Clark Kimberling, ck6@cedar.evansville.edu
%I A062174
%S A062174 0,1,0,3,1,3,1,3,0,3,1,3,1,3,9,11,1,9,1,7,9,3,1,3,6,3,0,27,1,3,1,11,9,
%T A062174 3,4,27,1,3,9,27,1,33,1,27,36,3,1,27,43,33,9,27,1,27,4,3,9,3,1,27,1,3,
%U A062174 9,43,16,45,1,27,9,13,1,27,1,3,69,27,25,9,1,27,0,3,1,75,81,3,9,75,1,63
%N A062174 3^(n-1) mod n.
%H A062174 Index entries for sequences related to pseudoprimes
%Y A062174 Cf. A062172.
%K A062174 nonn
%O A062174 1,4
%A A062174 Henry Bottomley (se16@btinternet.com), Jun 12 2001
%I A063062
%S A063062 1,1,1,1,1,1,1,1,1,1,3,1,3,1,3,1,3,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1,3,1,3,
%T A063062 1,1,1,1,1,1,1,1,1,1,1,3,1,3,1,3,1,3,1,1,1,1,1,1,1,1,1,1,0,3,1,3,1,3,1,
%U A063062 3,1,1,1,1,1,3,1,3,1,3,1,3,1,1,1,1,1,3,1,3,1,3,1,3,1,1,1,1,1,3,1,3,1,3
%N A063062 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063061 in order to obtain a term in the trajectory of 10553.
%e A063062 12097 is a term of A063061. One 'Reverse and Add!' operation applied to 12097 leads to a term (91118) in the trajectory of 10553, so the corresponding term of the present sequence is 1.
%Y A063062 A023108, A033865, A063060, A063061.
%K A063062 base,nonn
%O A063062 0,11
%A A063062 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jul 07 2001
%I A066056
%S A066056 1,1,1,3,1,3,1,3,1,3,1,3,1,1,3,1,3,1,1,1,1,3,3,1,3,1,3,1,3,1,1,3,1,3,1,
%T A066056 1,1,1,3,1,3,1,3,1,3,1,3,1,1,3,1,3,1,1,1,1,3,1,3,1,3,1,1,3,1,1,3,0,3,3,
%U A066056 3,3,3,3,1,3,1,3,3,3,3,3,3,1,3,1,3,3,3,3,3,3,1,3,1,3,2,2,2,2,3,2,3,2,1
%N A066056 Number of 'Reverse and Add!' operations that have to be applied to the n-th term of A066055 in order to obtain a term in the trajectory of 10583.
%e A066056 13597 is the fourth term of A066055. Three 'Reverse and Add!' operations applied to 13597 lead to a term (937838) in the trajectory of 10583, so the corresponding term of the present sequence is 3.
%Y A066056 Cf. A023108, A033865, A066054, A066055.
%K A066056 base,nonn
%O A066056 0,4
%A A066056 Klaus Brockhaus (klaus-brockhaus@t-online.de), Nov 30 2001
%I A010684
%S A010684 1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,
%T A010684 3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,
%U A010684 1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1
%N A010684 Period 2.
%K A010684 nonn
%O A010684 0,2
%A A010684 njas
%I A035652
%S A035652 0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,3,1,3,1,3,1,3,7,3,8,3,8,3,8,14,8,17,8,
%T A035652 18,8,18,26,18,33,18,36,18,37,47,37,61,37,68,37,71,81,72,106,72,121,
%U A035652 72,128,138,131,181,132,209,132,224,228,231,297,234,347,235,376,373
%N A035652 Partitions into parts 7k and 7k+2 with at least one part of each type
%K A035652 nonn,part
%O A035652 1,16
%A A035652 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A035689
%S A035689 0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,1,3,1,3,1,3,1,4,3,7,3,8,3,8,4,10,8,14,
%T A035689 9,16,9,18,11,22,17,28,19,32,21,36,25,44,35,52,40,60,44,68,52,82,66,
%U A035689 95,76,108,85,123,100,145,122,166,140,188,157,214,182,250,215,283,245
%N A035689 Partitions into parts 8k+2 and 8k+7 with at least one part of each type
%K A035689 nonn,part
%O A035689 1,17
%A A035689 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A050345
%S A050345 1,1,1,1,1,3,1,3,1,3,1,6,1,3,3,4,1,6,1,6,3,3,1,13,1,3,3,6,1,12,1,7,3,3,
%T A050345 3,15,1,3,3,13,1,12,1,6,6,3,1,25,1,6,3,6,1,13,3,13,3,3,1,31,1,3,6,12,3,
%U A050345 12,1,6,3,12,1,37,1,3,6,6,3,12,1,25,4,3,1,31,3,3,3,13,1,31,3,6,3,3
%N A050345 Ways to factor n into distinct factors with one level of parentheses.
%C A050345 Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
%C A050345 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).
%F A050345 Dirichlet g.f.: prod{n=2 to inf}(1+1/n^s)^A045778(n).
%e A050345 12=(12)=(6*2)=(6)*(2)=(4*3)=(4)*(3)=(3*2)*(2).
%Y A050345 Cf. A045778, A050346-A050350. a(p^k)=A050342. a(A002110)=A000258.
%K A050345 nonn
%O A050345 1,6
%A A050345 Christian G. Bower (bowerc@usa.net), Oct 1999.
%I A025810
%S A025810 1,0,1,0,1,1,1,1,1,1,3,1,3,1,3,3,3,3,3,3,6,3,6,3,6,6,6,
%T A025810 6,6,6,10,6,10,6,10,10,10,10,10,10,15,10,15,10,15,15,15,
%U A025810 15,15,15,21,15,21,15,21,21,21,21,21,21,28,21,28,21,28
%N A025810 Expansion of 1/((1-x^2)(1-x^5)(1-x^10)).
%K A025810 nonn
%O A025810 0,11
%A A025810 njas
%I A001319
%S A001319 1,0,1,0,1,1,1,1,1,1,3,1,3,1,3,3,3,3,3,3,7,3,7,3,7,7,7,
%T A001319 7,7,7,13,7,13,7,13,13,13,13,13,13,22,13,22,13,22,22,22,
%U A001319 22,22,22,35,22,35,22,35,35,35,35,35,35,53,35,53,35,53
%N A001319 Ways of making change for n cents using coins of 2, 5, 10, 20, 50 cents.
%D A001319 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D A001319 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%H A001319 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 184
%H A001319 Index entries for sequences related to making change.
%p A001319 1/(1-x^2)/(1-x^5)/(1-x^10)/(1-x^20)/(1-x^50)
%K A001319 nonn
%O A001319 0,11
%A A001319 njas
%I A046933
%S A046933 0,1,1,3,1,3,1,3,5,1,5,3,1,3,5,5,1,5,3,1,5,3,5,7,3,1,3,1,3,13,3,5,1,9,
%T A046933 1,5,5,3,5,5,1,9,1,3,1,11,11,3,1,3,5,1,9,5,5,5,1,5,3,1,9,13,3,1,3,13,5,
%U A046933 9,1,3,5,7,5,5,3,5,7,3,7,9,1,9,1,5,3,5,7,3,1,3,11,7,3,7,3,5,11,1,17
%N A046933 Number of composites between successive primes.
%C A046933 A046933(n) = A001223(n) - 1 for all n.
%F A046933 a(n) = p(n+1) - p(n) - 1
%e A046933 a(1) = 0 since 2 is adjacent to 3; a(2) = 1 since 4 is between 3 and 5; a(4) = 3 = 11 - 7 - 1, etc
%K A046933 easy,nonn,nice
%O A046933 1,4
%A A046933 Marc Le Brun (mlb@well.com)
%I A023511
%S A023511 3,1,3,1,3,7,3,5,3,3,1,19,3,11,3,3,3,31,17,3,37,5,3,3,7,3,13,3,5,3,
%T A023511 1,3,3,5,3,19,79,41,3,3,3,7,3,97,3,5,53,7,3,5,3,3,11,3,3,3,3,17,139,
%U A023511 3,71,3,7,3,157,3,83,13,3,5,3,3,23,11,5,3,3,199,3,5,3,211,3,7,5,3,3
%N A023511 Least odd prime divisor of p(n) + 1, or 1 if p(n) + 1 is a power of 2.
%K A023511 nonn
%O A023511 1,1
%A A023511 Clark Kimberling (ck6@cedar.evansville.edu)
%I A035628
%S A035628 0,0,0,0,0,0,1,0,1,0,1,3,1,3,1,3,7,3,8,3,8,14,8,17,8,18,26,18,33,18,
%T A035628 36,47,37,61,37,68,81,71,106,72,121,138,128,181,131,209,228,224,297,
%U A035628 231,347,372,376,482,391,566,592,619,760,648,898,934,989,1188,1043
%N A035628 Partitions into parts 5k and 5k+2 with at least one part of each type
%K A035628 nonn,part
%O A035628 1,12
%A A035628 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A057024
%S A057024 3,1,3,1,3,7,9,5,3,15,1,19,21,11,3,27,15,31,17,9,37,5,21,45,49,51,13,
%T A057024 27,55,57,1,33,69,35,75,19,79,41,21,87,45,91,3,97,99,25,53,7,57,115,
%U A057024 117,15,121,63,129,33,135,17,139,141,71,147,77,39,157,159,83,169,87
%N A057024 Largest odd factor of (n-th prime+1); k when n-th prime is written as k*2^m-1 [with k odd].
%H A057024 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime
%F A057024 a(n) =A000265(A000040(n)+1)) =A000265(A028815(n)) =(A000040(n)+1)/A007814(A000040(n)+1) =A028815(n)/A023512(n)
%e A057024 a(5)=3 because 5th prime is 11 and 11=3*2^2-1
%Y A057024 Cf. A057023.
%K A057024 nonn
%O A057024 1,1
%A A057024 Henry Bottomley (se16@btinternet.com), Jul 24 2000
%I A023892
%S A023892 1,3,1,3,1,3,8,5,1,23,1,9,1,32,46,75,50,3,1,117,92,135,1,79,274,257,
%T A023892 433,80,175,518,1146,757,859,218,1688,1539,3736,947,469,965,7463,
%U A023892 1186,8212,1191,8819
%V A023892 1,3,1,3,1,3,8,-5,1,23,1,-9,1,-32,46,75,-50,3,1,-117,92,135,1,79,-274,-257,
%W A023892 433,80,175,518,-1146,-757,859,-218,1688,1539,-3736,-947,469,-965,7463,
%X A023892 1186,-8212,1191,-8819
%N A023892 Derivative of log of A007360.
%K A023892 sign,done
%O A023892 1,2
%A A023892 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A035648
%S A035648 0,0,0,0,0,0,1,0,1,0,1,1,3,1,3,1,4,3,7,3,8,4,10,8,14,9,17,11,22,17,28,
%T A035648 20,34,25,43,35,53,42,64,51,80,67,96,80,115,98,142,123,168,147,200,
%U A035648 178,244,217,286,257,339,310,407,371,475,439,559,523,664,618,772,726
%N A035648 Partitions into parts 6k+2 and 6k+5 with at least one part of each type
%K A035648 nonn,part
%O A035648 1,13
%A A035648 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A053575
%S A053575 1,1,1,1,1,1,3,1,3,1,5,1,3,3,1,1,1,3,9,1,3,5,11,1,5,3,9,3,7,1,15,1,5,1,
%T A053575 3,3,9,9,3,1,5,3,21,5,3,11,23,1,21,5,1,3,13,9,5,3,9,7,29,1,15,15,9,1,3,
%U A053575 5,33,1,11,3,35,3,9,9,5,9,15,3,39,1,27,5,41,3,1,21,7,5,11,3,9,11,15,23
%N A053575 EulerPhi[n] is divisible by the maximal possible power of 2: the "odd kernel" of Phi[n].
%C A053575 This is not necessarily the square-free kernel. E.g. for n=19, Phi[19]=18 is divisible by 9, an odd square. Values at which this kernel is 1 correspond to A003401 (polygons constructible with ruler and compass)
%e A053575 n=70=2*5*7, Phi[70]=24=8*3, so the odd kernel of 70 a(70)=3
%Y A053575 A000010.
%K A053575 nonn
%O A053575 1,7
%A A053575 Labos E. (labos@ana1.sote.hu), Jan 18 2000
%I A013603
%S A013603 1,1,3,1,3,1,5,3,3,9,3,1,3,19,15,1,5,1,3,9,3,15,3,39,5,39,57,3,35,1,
%T A013603 5,9,41,31,5,25,45,7,87,21,11,57,17,55,21,115,59,81,27,129,47,111,
%U A013603 33,55,5,13,27,55,93,1,57,25,59,49,5,19,23,19,35,231,93,69,35,97,15
%N A013603 2^n-prevprime(2^n).
%H A013603 Table for large n
%p A013603 seq(2^i-prevprime(2^i),i=2..100);
%Y A013603 Cf. A014210.
%K A013603 nonn
%O A013603 2,3
%A A013603 James Kilfiger (mapdn@csv.warwick.ac.uk)
%I A050336
%S A050336 1,1,1,3,1,3,1,6,3,3,1,9,1,3,3,14,1,9,1,9,3,3,1,23,3,3,6,9,1,12,1,27,3,
%T A050336 3,3,31,1,3,3,23,1,12,1,9,9,3,1,57,3,9,3,9,1,23,3,23,3,3,1,41,1,3,9,58,
%U A050336 3,12,1,9,3,12,1,83,1,3,9,9,3,12,1,57,14,3,1,41,3,3,3,23,1,41,3,9
%N A050336 Ways of factoring n with one level of parentheses.
%C A050336 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).
%F A050336 Dirichlet g.f.: prod{n=2 to inf}(1/(1-1/n^s)^A001055(n)).
%e A050336 12=(12)=(6*2)=(6)*(2)=(4*3)=(4)*(3)=(3*2*2)=(3*2)*(2)=(3)*(2*2)=(3)*(2)*(2).
%Y A050336 Cf. A001055, A050337-A050341. a(p^k)=A001970. a(A002110)=A000258.
%K A050336 nonn
%O A050336 1,4
%A A050336 Christian G. Bower (bowerc@usa.net), Oct 1999.
%I A016572
%S A016572 3,1,3,1,8,14,1,3,1,9,2,9,22,59,11,1,2,3,2,5,3,3,4,1,7,
%T A016572 1,65,1,6,1,3,1,7,1,2,2,1,30,1,1,1,1,1,50,1,1,1,1,4,13,
%U A016572 13,4,1,23,1,2,2,8,1,2,2,3,2,2,3,1,1,1,8,1,4,1,1,3,1,9
%N A016572 Continued fraction for ln(89/2).
%K A016572 nonn,cofr
%O A016572 1,1
%A A016572 njas
%I A057741
%S A057741 1,1,1,3,1,3,2,1,5,0,2,1,5,0,0,4,1,7,2,0,0,2,1,7,0,0,0,0,6,1,9,0,2,0,0,
%T A057741 0,4,1,9,2,0,0,0,0,0,6,1,11,0,0,4,0,0,0,0,4,1,11,0,0,0,0,0,0,0,0,10,1,
%U A057741 13,2,2,0,2,0,0,0,0,0,4,1,13,0,0,0,0,0,0,0,0,0,0,12,1,15,0,0,0,0,6,0,0
%N A057741 Table T(n,k) giving number of elements of order k in dihedral group D_{2n} of order 2n, n >= 1, 1<=k<=g(n), where g(n) = 2 if n=1 else g(n) = n.
%C A057741 Note that D_2 equals the cyclic group of order 2.
%F A057741 If k<>2 and k does not divide n, this number is 0; if k<>2 and k divides n, this number is phi(k), where phi is the Euler totient function; if k=2, this number is n for odd n, and n+1 for even n.
%e A057741 1,1; 1,3; 1,3,2; 1,5,0,2; 1,5,0,0,4; ...
%Y A057741 Cf. A057731, A054522, A057740.
%K A057741 nonn,tabf,easy,nice
%O A057741 1,4
%A A057741 Roger CUCULIERE (cuculier@imaginet.fr), Oct 29 2000
%E A057741 More terms from James A. Sellers (sellersj@math.psu.edu), Oct 30 2000
%I A016571
%S A016571 3,1,3,2,2,60,1,26,15,1,1,4,1,1,2,2,3,1,1,1,5,1,8,2,1,1,
%T A016571 1,4,3,21,2,1,6,1,3,1,7,6,29,1,1,14,2,19,2,3,2,2,3,2,3,
%U A016571 1,1,1,1,4,3,1,7,1,2,4,17,5,1,3,14,2,97,1,8,3,1,9,1,5,12
%N A016571 Continued fraction for ln(87/2).
%K A016571 nonn,cofr
%O A016571 1,1
%A A016571 njas
%I A055189
%S A055189 3,1,3,2,3,1,1,3,3,3,1,1,2,6,3,5,1,2,2,7,3,6,1,4,2,1,6,1,5,8,3,9,1,5,2,
%T A055189 3,6,2,5,1,7,1,4,10,3,12,1,7,2,4,6,4,5,2,7,2,4,1,8,1,9,11,3,15,1,10,2,
%U A055189 5,6,5,5,4,7,5,4,2,8,2,9,1,10,1,12,12,3,18
%N A055189 Cumulative counting sequence: method A (adjective,noun)-pairs with 1st term 3.
%C A055189 Segments (as in %e): 3; 1,3; 2,3,1,1; 3,3,3,1,1,2; ...
%C A055189 Conjecture: every nonnegative integer occurs.
%e A055189 Write 3, thus having 1 3, thus having 2 3's and 1 1, thus having 3 3's and 3 1's and 1 2, etc.
%K A055189 nonn
%O A055189 1,1
%A A055189 Clark Kimberling, ck6@cedar.evansville.edu, Apr 27 2000
%I A059660
%S A059660 3,1,3,2,3,3,1,3,2,3,2,1,3,1,3,4,4,1,3,4,3,2,3,3,4,2,3,3,3,1,3,2,2,1,3,
%T A059660 1,3,4,4,1,2,1,3,4,2,3,3,4,3,2,2,1,3,1,3,2,2,1,3,1,3,3,4,2,2,1,3,4,5,2,
%U A059660 1,3,1,3,2,3,3,1,3,5,2,1,3,1,3,4,5,3,3,1,3,2,3,3,1,3,2,3,3,3,1,3,4,1,3
%N A059660 First differences of A059659.
%K A059660 nonn
%O A059660 1,1
%A A059660 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Feb 03 2001
%I A035456
%S A035456 0,1,0,1,1,1,1,1,1,3,1,3,2,3,4,3,4,5,4,8,5,8,8,8,11,11,11,15,12,19,17,
%T A035456 19,22,22,26,30,27,36,34,41,44,45,51,56,57,69,66,77,82,87,97,104,107,
%U A035456 125,124,143,149,158,175,186,196,221,221,252,263,280,307,321,344,379
%N A035456 Partitions into parts 8k+2 or 8k+5.
%K A035456 nonn,part
%O A035456 1,10
%A A035456 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A035664
%S A035664 0,0,0,0,0,0,1,0,1,0,1,1,1,3,1,3,2,3,4,3,7,4,8,6,8,10,9,15,11,17,15,
%T A035664 19,22,21,29,25,35,33,39,44,44,57,52,67,65,76,84,86,103,101,122,124,
%U A035664 139,153,158,185,185,216,222,247,268,282,317,327,369,387,422,458,482
%N A035664 Partitions into parts 7k+2 and 7k+5 with at least one part of each type
%K A035664 nonn,part
%O A035664 1,14
%A A035664 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A035557
%S A035557 1,0,1,0,3,1,3,2,6,7,8,9,14,18,26,24,38,41,65,68,88,104,137,170,204,237,
%T A035557 302,361,464,518,645,758,949,1120,1320,1562,1888,2250,2688,3092,3717,
%U A035557 4349,5228,6054,7096,8308,9803,11488,13380,15468,18153,21080,24693,28374
%N A035557 Partitions of n with equal number of parts congruent to each of 1 and 3 (mod 5)
%K A035557 nonn,part
%O A035557 0,5
%A A035557 Olivier Gerard (ogerard@ext.jussieu.fr)
%E A035557 More terms from dww
%I A058589
%S A058589 1,3,1,3,2,9,2,9,1,24,0,27,5
%V A058589 1,-3,-1,-3,-2,-9,2,-9,-1,-24,0,-27,5
%N A058589 McKay-Thompson series of class 24f for Monster.
%D A058589 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058589 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058589 sign,done
%O A058589 -1,2
%A A058589 njas, Nov 27, 2000
%I A050141
%S A050141 3,1,3,3,1,3,1,3,3,1,3,3,1,3,1,3,3,1,3,1,3,3,1,3,3,1,3,1,3,3,1,3,3,1,3,
%T A050141 1,3,3,1,3,1,3,3,1,3,3,1,3,1,3,3,1,3,1,3,3,1,3,3,1,3,1,3,3,1,3,3,1,3,1,
%U A050141 3,3,1,3,1,3,3,1,3,3,1,3,1,3
%N A050141 a(n)=b(n+1)-b(n), where b=A050140 (last digit in repeating block in continued fraction of n*tau).
%C A050141 The substitutions 3->1 and 1->0 carry this sequence onto the infinite Fibonacci word (A005614).
%K A050141 nonn
%O A050141 1,1
%A A050141 Clark Kimberling, ck6@cedar.evansville.edu
%I A050306
%S A050306 1,3,1,3,3,1,5,0,3,1,8,8,0,3,1,5,0,5,0,3,1,1,8,3,5,0,3,1,5,9,0,0,5,0,3,
%T A050306 1,17,3,15,3,0,5,0,3,1,6,4,2,7,0,0,5,0,3,1,1,15,0,8,4,0,0,5,0,3,1,14,
%U A050306 20,1,2,0,7,0,0,5,0,3,1,11,2,19,0,8,3
%V A050306 1,-3,1,3,-3,1,5,0,-3,1,-8,8,0,-3,1,-5,0,5,0,-3,1,-1,-8,3,5,0,-3,1,5,
%W A050306 -9,0,0,5,0,-3,1,17,-3,-15,3,0,5,0,-3,1,6,4,-2,-7,0,0,5,0,-3,1,-1,15,
%X A050306 0,-8,-4,0,0,5,0,-3,1,-14,20,1,-2,0,-7,0,0,5,0,-3,1,-11,2,19,0,-8,3
%N A050306 Matrix cube of inverse partition triangle A038498.
%e A050306 1; -3,1; 3,-3,1; 5,0,-3,1; ...
%Y A050306 Cf. A050305-A050313.
%K A050306 sign,done,tabl
%O A050306 1,2
%A A050306 Christian G. Bower (bowerc@usa.net), Aug 1999.
%I A059787
%S A059787 1,1,1,3,1,3,3,3,1,1,5,5,1,3,3,1,9,5,3,7,3,5,1,1,3,9,5,9,5,1,3,1,3,3,9,
%T A059787 5,3,5,3,1,1,5,1,3,3,3,9,3,3,3,1,1,5,1,7,15,3,5,3,1,3,1,3,9,5,7,11,3,7,
%U A059787 3,3,1,5,5,3,3,9,3,7,3,1,11,1,11,3,1,9,5,7,3,3,9,3,1,11,3,1,7,3,5,3,3
%N A059787 Distance between 2*(nth prime) and next prime.
%e A059787 If a(n)=1 then Prime[n] is Sophie Germain prime.
%p A059787 with(numtheory): [seq(nextprime(2*ithprime(k))-2*ithprime(k),k=1..256)];
%Y A059787 A005385, A005384.
%K A059787 nonn
%O A059787 1,4
%A A059787 Labos E. (labos@ana1.sote.hu), Feb 22 2001
%I A059789
%S A059789 1,1,3,1,3,3,3,1,3,5,1,1,3,3,5,3,5,9,3,3,7,1,3,5,1,3,7,3,7,3,3,5,3,1,5,
%T A059789 9,1,9,3,9,5,3,3,3,5,1,1,3,5,1,3,11,3,3,5,3,15,1,7,5,3,9,1,3,7,3,1,1,3,
%U A059789 7,5,9,1,3,1,5,5,7,5,7,9,3,3,3,1,3,11,3,3,7,5,5,3,5,1,9,5,3,7,13,1,5,3
%N A059789 Distance of 2*Prime[n] from previous prime.
%p A059789 with(numtheory): [seq(2*ithprime(k)-prevprime(2*ithprime(k)),k=1..256)];
%Y A059789 A005382-A005385.
%K A059789 nonn
%O A059789 0,3
%A A059789 Labos E. (labos@ana1.sote.hu), Feb 22 2001
%I A023136
%S A023136 1,1,3,1,3,3,3,1,5,3,3,3,3,3,9,1,5,5,3,3,9,3,3,3,5,3,7,3,3,9,7,1,9,5,9,5,
%T A023136 3,3,9,3,5,9,7,3,15,3,3,3,5,5,15,3,3,7,9,3,9,3,3,9,3,7,23,1,13,9,3,5,9,9,
%U A023136 3,5,9,3,15,3,9,9,3,3,9,5,3,9,23,7,9,3,9,15,17,3,21,3,9,3,5,5,15,5,3,15
%N A023136 Number of cycles of function f(x) = 4x mod n.
%K A023136 nonn
%O A023136 1,3
%A A023136 dww
%I A063195
%S A063195 0,1,1,1,1,3,1,3,3,3,3,5,3,5,5,5,5,7,5,7,7,7,7,9,7,9,9,9,9,11,9,
%T A063195 11,11,11,11,13,11,13,13,13,13,15,13,15,15,15,15,17,15,17
%N A063195 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 6 ).
%H A063195 William A. Stein (was@math.berkeley.edu), Dimensions of the spaces S_k^{new}(Gamma_0(N))
%H A063195 William A. Stein (was@math.berkeley.edu), The modular forms database
%K A063195 nonn
%O A063195 1,6
%A A063195 njas, Jul 10 2001
%I A025796
%S A025796 1,0,1,1,1,1,3,1,3,3,3,3,6,3,6,6,6,6,10,6,10,10,10,10,15,
%T A025796 10,15,15,15,15,21,15,21,21,21,21,28,21,28,28,28,28,36,
%U A025796 28,36,36,36,36,45,36,45,45,45,45,55,45,55,55,55,55,66
%N A025796 Expansion of 1/((1-x^2)(1-x^3)(1-x^6)).
%K A025796 nonn
%O A025796 0,7
%A A025796 njas
%I A024163
%S A024163 0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,3,1,3,3,3,3,6,3,6,6,6,6,10,6,10,10,10,10,
%T A024163 15,10,15,15,15,15,21,15,21,21,21,21,28,21,28,28,28,28,36,28,36,36,36,36,45,
%U A024163 36,45,45,45,45,55,45,55,55,55,55,66,55,66,66,66,66,78,66,78,78,78,78,91
%N A024163 Number of integer-sided triangles with sides a,b,c, aIndex entries for sequences related to f.c.c. lattice
%K A005885 nonn,easy
%O A005885 0,1
%A A005885 njas
%I A061892
%S A061892 0,3,1,3,3,6,10,28,108,1011,32511
%N A061892 Lionel-Levine-sequence generated by (1,0,0).
%C A061892 A011784 allows the addition of a(-1)=0 and a(0)=2 using row A012257(-2)=1 and A012257(-1)=0 resp. row A12257(0)=2. In this sense A011784 / A012257 are "generated" by (1,0), A061892 / A061893 by (1,0,0), A061894 / A061895 by (2,0).
%e A061892 a(4) = 3: (1,0,0),(3),(1,1,1),(1,2,3),(1,1,1,2,2,3)
%o A061892 (REXX) /* replace 1 0 0 by 1 0 to get A011784 */
%o A061892 S = ''; A = 1 0 0; do N = 1 to 10; T = words( A );
%o A061892 S = S word( A, T ); B = A; A = ''; do K = 1 to T;
%o A061892 A = A space( copies( K '', word( B, T + 1 - K )));
%o A061892 end K; end N; T = words( A ); say S word( A, T ) T;
%Y A061892 A011784, A012257, A061893, A061894.
%K A061892 nonn,easy,huge,more
%O A061892 0,2
%A A061892 Frank.Ellermann@t-online.de, May 13 2001
%I A038573
%S A038573 0,1,1,3,1,3,3,7,1,3,3,7,3,7,7,15,1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31,
%T A038573 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31,3,7,7,15,7,15,15,31,7,15,15,31,
%U A038573 15,31,31,63,1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31,3,7,7,15,7,15,15,31
%N A038573 Smallest number with same number of 1's in its binary expansion as n.
%F A038573 a(2n) = a(n), a(2n+1) = 2*a(n)+1, a(0) = 0. a(n) = A001316(n)-1 = 2^A000120(n)-1 (comment from Daniele Parisse (daniele.parisse@m.dasa.de)).
%e A038573 9 = 1001 -> 0011 -> 3, so a(9)=3.
%t A038573 Array[ 2^Count[ IntegerDigits[ #,2 ],1 ]-1&,100 ]
%o A038573 (PARI) a(n)=2^subst(Pol(binary(n)),x,1)-1
%K A038573 nonn,easy,nice
%O A038573 0,4
%A A038573 Marc Le Brun (mlb@well.com)
%E A038573 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A030708
%S A030708 1,1,1,3,1,3,4,1,2,3,4,6,1,2,3,4,6,8,1,2,3,4,5,6,8,11,1,2,3,4,
%T A030708 5,6,8,11,13,1,2,3,4,5,6,7,8,10,11,13,16,1,2,3,4,5,6,7,8,9,10,
%U A030708 11,12,13,16,18,1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,18,22,1,2
%N A030708 Row 2, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in rows 1 and 2, for j=1,2,...,m, where m=max number in row 1 for stages 1 to j-1; stage 1 is 1 in row 1.
%Y A030708 A030707=row 1
%K A030708 nonn
%O A030708 1,4
%A A030708 Clark Kimberling, ck6@cedar.evansville.edu
%I A064884
%S A064884 3,1,3,4,1,3,7,4,5,1,3,10,7,11,4,9,5,6,1,3,13,10,17,7,18,11,15,4,13,9,
%T A064884 14,5,11,6,7,1,3,16,13,23,10,27,17,24,7,25,18,29,11,26,15,19,4,17,13,
%U A064884 22,9,23,14,19,5,16,11,17,6,13
%N A064884 Eisenstein array Ei(3,1).
%C A064884 In Eisenstein's notation this is the array for m=3 and n=1; see pp.41-2 of the Eisenstein ref. given for A064881. This is identical with the array for m=1,n=3, given in A064883, read backwards. The array for m=n=1 is A049456.
%C A064884 For n >= 1, the number of entries of row n is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 4*A007051(n-1).
%C A064884 The binary tree built from the rationals a(n,m)/a(n,m+1), m=0..2^(n-1), for each row n >= 1 gives the sub-tree of the (Eisenstein-)Stern-Brocot tree in the version of, e.g., Calkin and Wilf (for the ref. see A002487, also for the Wilf link) with root 3/1. The composition rule of this tree is i/j -> i/(i+j), (i+j)/j.
%F A064884 a(n,m)= a(n-1,m/2) if m is even, else a(n,m)= a(n-1,(m-1)/2)+a(n-1,(m+1)/2, a(1,0)=3, a(1,1)=1.
%e A064884 {3,1}; {3,4,1}; {3,7,4,5,1}; {3,10,7,11,4,9,5,6,1}; ...
%e A064884 This binary subtree of rationals is built from 3/1; 3/4,4/1; 3/7,7/4,4/5,5/1; ...
%K A064884 nonn,easy,tabf
%O A064884 1,1
%A A064884 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Oct 19 2001
%I A027011
%S A027011 1,3,1,3,4,1,3,7,5,1,3,7,15,6,1,3,7,18,28,7,1,3,7,18,44,47,8,
%T A027011 1,3,7,18,47,98,73,9,1,3,7,18,47,120,199,107,10,1,3,7,18,47,123,
%U A027011 291,373,150,11,1,3,7,18,47,123,319,661,654,203,12,1,3,7,18,47
%N A027011 Triangular array T by rows: T(n,k)=t(n,2k+1) for 0<=k<=[ (2n-1)/2 ], T(n,n)=1, t given by A027960, n >= 0.
%Y A027011 This is a bisection of the "Lucas array " A027960; see A026998 for the other bisection.
%K A027011 nonn,tabl
%O A027011 1,2
%A A027011 Clark Kimberling, ck6@cedar.evansville.edu
%I A008311
%S A008311 1,1,1,1,3,1,3,4,1,10,5,1,10,15,6,1,64,35,21,7,1,35,56,28,8,1,126,
%T A008311 84,36,9,1,126,210,120,45,10,1,462,330,165,55,11,1,462,792,495,220,
%U A008311 66,12,1
%N A008311 Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x).
%D A008311 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
%H A008311 Index entries for sequences related to Chebyshev polynomials.
%K A008311 nonn,tabl
%O A008311 0,5
%A A008311 njas
%I A050121
%S A050121 3,1,3,4,3,4,4,7,4,3,11,7,4,18,11,3,4,29,18,4,7,47,29,4,3,11,76,47,7,4,
%T A050121 18,123,76,11,3,4,29,199,123,18,4,7,47,322,199,29,4,3,11
%N A050121 a(n)=b(n)-b(n-1), where b=A050120 (ordered products of Fibonacci and Lucas numbers).
%K A050121 nonn
%O A050121 2,1
%A A050121 Clark Kimberling, ck6@cedar.evansville.edu
%I A029152
%S A029152 1,0,1,1,1,1,3,1,3,4,3,4,7,4,7,9,7,9,14,9,14,17,14,17,24,
%T A029152 17,24,29,24,29,38,29,38,45,38,45,57,45,57,66,57,66,81,
%U A029152 66,81,93,81,93,111,93,111,126,111,126,148,126,148,166
%N A029152 Expansion of 1/((1-x^2)(1-x^3)(1-x^6)(1-x^9)).
%K A029152 nonn
%O A029152 0,7
%A A029152 njas
%I A008924
%S A008924 1,0,0,0,0,0,1,0,1,1,1,1,3,1,3,4,5,5,10,8,14,17,22,27,44,
%T A008924 45,69,90,123,157,240,299,445,610,884,1237,1871,2635,3975,
%U A008924 5823,8751,12918,19570,28977,43644,64880,96983,143510,213118
%N A008924 Harmonic Molien series for Conway group Con.0.
%D A008924 W. C. Huffman and N. J. A. Sloane, "Most Primitive Groups have Messy Invariants", Advances in Math. 32 (1979) 118-127.
%K A008924 nonn
%O A008924 0,13
%A A008924 njas,jhc (Aug 14, 1977)
%I A021323
%S A021323 0,0,3,1,3,4,7,9,6,2,3,8,2,4,4,5,1,4,1,0,6,5,8,3,0,7,2,1,0,0,3,1,3,
%T A021323 4,7,9,6,2,3,8,2,4,4,5,1,4,1,0,6,5,8,3,0,7,2,1,0,0,3,1,3,4,7,9,6,2,
%U A021323 3,8,2,4,4,5,1,4,1,0,6,5,8,3,0,7,2,1,0,0,3,1,3,4,7,9,6,2,3,8,2,4,4
%N A021323 Decimal expansion of 1/319.
%K A021323 nonn,cons
%O A021323 0,3
%A A021323 njas
%I A005474 M2215
%S A005474 1,1,1,1,1,3,1,3,5,3,3,7,3,5,7,3,3,5,9,7,3,5,5,15,9,19,5,13,9,9,5,19,9,
%T A005474 5,7,15,13,9,9,15,25,13,9,27,19,15,21,7,13,11,23,9,13,13,11,33,15,25,23,15,13
%N A005474 Class numbers of quadratic fields.
%D A005474 D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
%H A005474 Index entries for sequences related to quadratic fields
%K A005474 nonn
%O A005474 1,6
%A A005474 njas
%I A012264
%S A012264 1,3,1,3,5,3,5,7,5,7,5,7,9,7,9,7,9,11,9,11,13,11,13,11,13,15,13,15,17,
%T A012264 15,17,15,17,19,17,19,17,19,21,19,21,23,21,23,21,23,25,23,25,23,25,27,
%U A012264 25,27,29,27,29,27,29,31,29,31,33
%N A012264 Number of real roots of x - x^3/3! + x^5/5! - ... + (-1)^n*x^(2n+1)/(2n+1)!.
%D A012264 Jim Propp (propp@math.mit.edu), posting to math-fun mailing list May 30 1997.
%p A012264 Digits:=25: t1:=0: for k from 1 by 2 to 51 do t1:=t1+(-1)^( (k-1)/2 )*x^k/k!; print(nops([ fsolve(t1*k!) ])); od:
%Y A012264 Cf. A012265.
%K A012264 nonn,nice
%O A012264 0,2
%A A012264 njas
%E A012264 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A063198
%S A063198 0,1,3,1,3,5,3,5,7,5,7,9,7,9,11,9,11,13,11,13,15,13,15,17,15,17,
%T A063198 19,17,19,21,19,21,23,21,23,25,23,25,27,25,27,29,27,29,31,29,31,
%U A063198 33,31,33
%N A063198 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 10 ).
%H A063198 William A. Stein (was@math.berkeley.edu), Dimensions of the spaces S_k^{new}(Gamma_0(N))
%H A063198 William A. Stein (was@math.berkeley.edu), The modular forms database
%K A063198 nonn
%O A063198 1,3
%A A063198 njas, Jul 10 2001
%I A016471
%S A016471 3,1,3,5,3,36,3,2,3,1,2,3,2,1,4,1,1,1,2,5,12,96,2,1,2,2,
%T A016471 2,1,1184,2,3,5,1,16,1,4,1,2,3,1,1,1,1,23,1,2,2,1,32,1,
%U A016471 1,7,1,2,8,1,1,1,1,1,1,5,6,1,3,1,4,1,1,1,5,2,1,17,42,2
%N A016471 Continued fraction for ln(43).
%K A016471 nonn,cofr
%O A016471 1,1
%A A016471 njas
%I A016646
%S A016646 3,1,3,5,4,9,4,2,1,5,9,2,9,1,4,9,6,9,0,8,0,6,7,5,2,8,3,1,8,1,0,1,9,
%T A016646 6,1,1,8,4,4,2,3,8,0,3,1,4,8,4,0,4,3,5,7,4,1,9,9,8,6,3,5,3,7,7,4,8,
%U A016646 2,9,9,3,2,4,5,9,8,4,7,9,8,2,9,8,1,9,8,4,0,1,0,9,2,1,5,2,9,9,4,8,1
%N A016646 Decimal expansion of ln(23).
%D A016646 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
%K A016646 nonn,cons
%O A016646 1,1
%A A016646 njas
%I A006257 M2216
%S A006257 0,1,1,3,1,3,5,7,1,3,5,7,9,11,13,15,1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,
%T A006257 31,1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55
%N A006257 Josephus problem.
%D A006257 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 10.
%H A006257 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A006257 Write n in binary, rotate left 1 place.
%Y A006257 Cf. A038572, A053644, A053645.
%Y A006257 a(n) = 2 * A053645(n) + 1 = 2(n-msb(n))+1. - Marc Le Brun (mlb@well.com), Jul 11 2001
%K A006257 nonn,easy,nice
%O A006257 0,4
%A A006257 njas
%I A050820
%S A050820 1,1,3,1,3,5,11,1,5,1,5,3,3,7,1,29,7,25,11,9,11,3,11,5,9,5,17,9,7,1,41,
%T A050820 7,11,7,23,15,21,11,1,13,57,5,25,15,9,9,45,29,35,35,65,11,15,3,67,13,9,
%U A050820 5,81,9,33,11,23,9,53,1,21,99,57,55,81,61,73,43,45,65,127,9,89,9,9,75
%N A050820 Odd numbers in the sequence generated by a(n)=|a(n-1)+2a(n-2)-n|.
%Y A050820 Cf. A005210.
%K A050820 nonn,easy
%O A050820 1,3
%A A050820 Mohammad K. Azarian, azarian@evansville.edu
%E A050820 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A049324
%S A049324 1,3,1,3,6,1,0,15,9,1,0,18,36,12,1,0,9,81,66,15,1,0,0,108,216,105,18,1,
%T A049324 0,0,81,459,450,153,21,1,0,0,27,648,1305,810,210,24,1,0,0,0,594,2673,
%U A049324 2970,1323,276,27,1,0,0,0,324,3915,7938
%N A049324 A convolution triangle of numbers generalizing Pascal's triangle A007318.
%H A049324 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A049324 a(n,m) = 3*(3*m-n+1)*a(n-1,m)/n + m*a(n-1,m-1)/n, n >= m >= 1; a(n,m):=0, nTransforms
%e A055885 1; 1,3; 1,3,6; 1,6,9,14; 1,6,18,23,27; ...
%Y A055885 Row sums give A007713. Cf. A055884, A055886.
%K A055885 nonn,tabl
%O A055885 1,3
%A A055885 Christian G. Bower (bowerc@usa.net), Jun 09 2000
%I A033789
%S A033789 1,1,1,3,1,3,6,3,6,8,7,10,13,9,11,20,18,19,27,19,27,45,
%T A033789 27,35,47,39,54,61,45,48,83,76,63,95,72,86,145,92,101,143,
%U A033789 119,145,170,126,136,216,193,160,239,187,191
%N A033789 Product t2(q^d); d | 30, where t2 = theta2(q)/(2*q^(1/4)).
%K A033789 nonn
%O A033789 0,4
%A A033789 njas
%I A058659
%S A058659 1,0,3,1,3,6,6,9,15,15,21,30,34,42,60,66,84,108,127,153,201,226,276,
%T A058659 342,400
%N A058659 McKay-Thompson series of class 39A for Monster.
%D A058659 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058659 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058659 nonn
%O A058659 -1,3
%A A058659 njas, Nov 27, 2000
%I A053642
%S A053642 1,1,3,1,3,6,7,1,3,6,7,12,13,14,15,1,3,6,7,12,13,14,15,24,25,26,27,28,
%T A053642 29,30,31,1,3,6,7,12,13,14,15,24,25,26,27,28,29,30,31,48,49,50,51,52,
%U A053642 53,54,55,56,57,58,59,60,61,62,63,1
%N A053642 Rotate one binary digit to the left, calculate, then rotate one binary digit to the right.
%F A053642 a(n) = A038572(A006257(n)), =n if 3*2^(k-1)<=n<2^(k+1), =a(n-2^(k-1)) if 2^k<=n<3*2^(k-1)
%e A053642 a(22)=14 because starting with 10110 the left rotation produces 01101 written as 1101 (i.e. 13) and the left rotation produces 1110 (i.e. 14)
%Y A053642 Cf. A006257, A038572.
%K A053642 nonn
%O A053642 1,3
%A A053642 Henry Bottomley (se16@btinternet.com), Mar 22 2000
%I A035619
%S A035619 0,0,0,0,1,0,1,3,1,3,7,3,8,14,8,17,26,18,33,47,36,61,81,68,106,137,
%T A035619 121,181,224,209,296,362,347,478,570,565,750,890,894,1166,1360,1396,
%U A035619 1774,2062,2134,2677,3076,3228,3973,4555,4804,5854,6657,7085,8513
%N A035619 Partitions into parts 3k and 3k+2 with at least one part of each type
%K A035619 nonn,part
%O A035619 1,8
%A A035619 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A064434
%S A064434 0,1,0,1,3,1,3,7,6,3,7,3,7,1,3,7,15,13,8,17,14,7,15,7,15,5,11,23,18,7,
%T A064434 15,31,30,27,20,5,11,23,8,17,35,29,16,33,22,45,44,41,34,19,39,27,2,5,
%U A064434 11,23,47,37,16,33,6,13,27,55,46,27,55,43,18,37,4,9,19,39,4,9,19,39,0
%N A064434 a(n) = remainder of (2*a(n-1) + 1) when divided by n.
%C A064434 Can be generalised to a(n) = f(a(n-1)) mod n, where f is any polynomial function.
%F A064434 a(n) = (a(n-1) * 2 + 1 ) mod n
%e A064434 0, (0*2+1) mod 2 = 1, (1*2+1) mod 3 = 0, (0*2+1) mod 4 = 1, (1*2+1) mod 5 = 3 (3*2+1) mod 6 =1
%Y A064434 Cf. A064456.
%K A064434 nonn
%O A064434 1,5
%A A064434 Jonathan Ayres (JonathanAyres@btinternet.com), Oct 01 2001
%I A019603
%S A019603 1,3,1,3,8,0,5,2,6,5,0,2,6,7,0,2,6,2,5,4,5,5,9,3,0,9,0,3,0,0,7,1,6,
%T A019603 5,3,0,6,9,2,8,4,4,4,3,9,0,1,1,7,7,0,9,9,9,4,1,3,5,3,2,3,2,5,0,7,9,
%U A019603 6,8,0,4,4,0,1,7,1,6,5,1,2,4,3,1,9,3,4,2,4,0,5,9,5,3,4,0,0,7,2,9,1
%N A019603 Decimal expansion of 2*Pi*E/13.
%K A019603 nonn,cons
%O A019603 1,2
%A A019603 njas
%I A046544
%S A046544 1,1,3,1,3,8,1,10,3,110,3,406
%N A046544 First denominator and then numerator of the central elements of the 1/3-Pascal triangle (by row).
%e A046544 1/1; 1/1 1/1; 1/1 1/3 1/1; 1/1 4/3 4/3 1/1; 1/1 7/3 8/3 7/3 1/1; 1/1 10/3 5/1 5/1 10/3 1/1; 1/1 13/3 25/3 10/1 25/3 13/3 1/1; 1/1 16/3 38/3 55/3 55/3 38/3 16/3 1/1; ...
%Y A046544 Cf. A046534.
%K A046544 tabl,nonn
%O A046544 1,3
%A A046544 Mohammad K. Azarian, ma3@cedar.evansville.edu
%I A011088
%S A011088 3,1,3,8,2,8,8,9,9,2,7,1,4,9,9,6,0,8,0,4,5,5,7,7,2,1,2,7,2,8,7,0,1,
%T A011088 2,3,7,3,5,1,2,0,7,2,3,8,5,2,5,1,7,9,4,6,1,6,5,0,9,9,9,8,7,5,7,5,2,
%U A011088 3,5,3,1,9,0,5,9,5,3,8,4,6,3,3,2,3,2,3,0,2,5,0,5,2,4,4,6,6,0,6,5,3
%N A011088 Decimal expansion of 4th root of 97.
%K A011088 nonn,cons
%O A011088 1,1
%A A011088 njas
%I A010282
%S A010282 3,1,3,9,1,2,3,1,6,1,1,8,1,1,1,7,2,9,6,1,6,1,26,3,7,2,1,
%T A010282 48,1,1,1,9,1,5,2,1,2,1,17,6,2,1,1,1,4,3,2,4,1,1,2,2,4,
%U A010282 1,1,4,1,1,2,1,8,1,1,1,35,1,1,1,1,1,3,17,1,1,4,1,3,2,2
%N A010282 Continued fraction for cube root of 53.
%K A010282 nonn,cofr
%O A010282 0,1
%A A010282 njas
%I A037095
%S A037095 1,1,3,1,3,9,11,17,19,25,123,65,195,169,171,753,435,249,2267,4065,8163,
%T A037095 841,843,31313,29651
%N A037095 Descending Binary Diagonal transformation of powers of 3 (A000244).
%F A037095 The transformation is: b[ n ] := Sum(bit_n(a[ n-i ],i)*(2^i),i=0..(n-1)) [ bit_n := (x,n) -> `mod`(floor(x/(2^n)),2); ]
%Y A037095 A000244, A037093 (relative to Fib Ascending Diagonal)
%K A037095 nonn
%O A037095 0,3
%A A037095 Antti Karttunen (karttu@megabaud.fi), 28.Jan, 1999
%I A058842
%S A058842 1,3,1,3,9,27,81,243,217,651,1953,1763,5289,15867,14833,44499,2425,
%T A058842 7275,21825,65475,196425,589275,1767825,5303475,15910425,47731275,
%U A058842 8976097,26928291,80784873,242354619,727063857,2181191571,6543574713
%N A058842 From Renyi's "beta expansion of 1 in base 3/2": sequence gives a(1), a(2), ... where x(n) = a(n)/2^n, with 0 < a(n) < 2^n, a(1) = 1, a(n) = 3*a(n-1) modulo 2^n.
%C A058842 Let r be a real number strictly between 1 and 2, x any real number between 0 and 1; define y = (y(i)) by x(0) = x; x(i+1) = r*x(i)-1 if r*x(i)>1, and r*x(i) otherwise; y(i) = integer part of x(i+1): y = (y(i)) is an infinite word on the alphabet (0,1). Here we take r = 3/2 and x = 1.
%D A058842 A. Renyi (1957), Representation for real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hung., 8, 477-493.
%Y A058842 Cf. A058841, A058840.
%K A058842 nonn,nice,easy
%O A058842 1,2
%A A058842 Claude Lenormand (hlen.lenormand@voonoo.net), Jan 05 2001
%E A058842 More terms from Larry Reeves (larryr@acm.org), Feb 22 2001
%I A025238
%S A025238 3,1,3,10,36,137,543,2219,9285,39587,171369,751236,3328218,14878455,
%T A025238 67030785,304036170,1387247580,6363044315,29323149825,135700543190,
%U A025238 630375241380,2938391049395,13739779184085,64430797069375,302934667061301
%N A025238 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.
%F A025238 G.f.: (1+3*x-sqrt(1-6*x+5*x^2))/2 - Michael Somos, June 8, 2000.
%o A025238 (PARI) a(n)=polcoeff((1+3*x-sqrt(1-6*x+5*x^2+x*O(x^n)))/2,n)
%Y A025238 Essentially same as A002212.
%K A025238 nonn
%O A025238 1,1
%A A025238 Clark Kimberling (ck6@cedar.evansville.edu)
%I A001351 M2217 N0879
%S A001351 0,1,3,1,3,11,9,8,27,37,33,67,117,131,192,341,459,613,999,1483,2013,
%T A001351 3032,4623,6533,9477,14311,20829,30007,44544,65657,95139,139625,206091
%N A001351 Associated Mersenne numbers.
%D A001351 C. B. Haselgrove, Associated Mersenne numbers, Eureka, 11 (1949), 19-22.
%Y A001351 Cf. A001350.
%K A001351 nonn,easy
%O A001351 0,3
%A A001351 njas, rkg
%I A040173
%S A040173 1,3,1,3,19,53,103,4111,78293
%N A040173 Numerator of probability that 2 elements of S_n chosen at random (with replacement) generate S_n.
%H A040173 E. W. Weisstein, Link to a section of The World of Mathematics.
%e A040173 Values for n=1,2,3,... are 1, 3/4, 1/2, 3/8, 19/40, ...
%Y A040173 Probability is A040173/A040174 = A040175/n!.
%K A040173 nonn,frac
%O A040173 1,2
%A A040173 Dan Hoey (Hoey@aic.nrl.navy.mil)
%I A021765
%S A021765 0,0,1,3,1,4,0,6,0,4,4,6,7,8,0,5,5,1,9,0,5,3,8,7,6,4,7,8,3,1,8,0,0,
%T A021765 2,6,2,8,1,2,0,8,9,3,5,6,1,1,0,3,8,1,0,7,7,5,2,9,5,6,6,3,6,0,0,5,2,
%U A021765 5,6,2,4,1,7,8,7,1,2,2,2,0,7,6,2,1,5,5,0,5,9,1,3,2,7,2,0,1,0,5,1,2
%N A021765 Decimal expansion of 1/761.
%K A021765 nonn,cons
%O A021765 0,4
%A A021765 njas
%I A051512
%S A051512 3,1,4,0,226,0,3,0,1,0,1,1,0,2,1,1,1,3,1,4,0,0,3,1,1,0,1,0,2,1,
%T A051512 4,2,3,0,4,0,1,1,0,0,1,3,1,4,2,1,0,1,1,1,0,0,2,1,1,2,3,1,0,5,0,
%U A051512 1,2,1,3,1,2,1,6,0,2,1,1,2,3,1,1,2,3,0,2,0,0,1,2,0,1,9,2,2,2,10
%V A051512 -3,-1,-4,0,-226,0,3,0,1,0,-1,-1,0,-2,-1,-1,-1,3,1,-4,0,0,3,1,-1,0,-1,0,-2,-1,
%W A051512 4,-2,-3,0,4,0,-1,-1,0,0,-1,-3,-1,4,-2,1,0,-1,1,-1,0,0,-2,-1,-1,-2,-3,1,0,5,0,
%X A051512 1,-2,-1,-3,-1,2,1,6,0,2,1,-1,-2,-3,-1,-1,2,-3,0,2,0,0,-1,-2,0,-1,9,-2,2,-2,10
%N A051512 [tan(n-th prime)].
%K A051512 sign,done
%O A051512 1,1
%A A051512 njas
%I A055187
%S A055187 1,1,1,3,1,4,1,1,3,6,1,2,3,1,4,8,1,3,3,2,4,1,6,1,2,11,1,5,3,3,4,2,6,3,
%T A055187 2,1,8,13,1,8,3,4,4,3,6,5,2,2,8,1,11,1,5,16,1,10,3,6,4,4,6,7,2,4,8,2,
%U A055187 11,3,5,1,13,18,1,12,3,9,4,6,6,9,2,5,8,3,11,4
%N A055187 Cumulative counting sequence: method A (adjective,noun)-pairs with 1st term 1.
%C A055187 Segments (as in %e): 1; 1,1; 3,1; 4,1,1,3; 6,1,2,3,1,4; ...
%C A055187 Conjecture: every nonnegative integer occurs.
%e A055187 Write 1, thus having 1 1, thus having 3 1's, thus having 4 1's and 1 3, etc.
%K A055187 nonn
%O A055187 1,4
%A A055187 Clark Kimberling, ck6@cedar.evansville.edu, Apr 27 2000
%I A010285
%S A010285 3,1,4,1,2,1,7,1,2,9,1,2,10,1,7,2,1,6,5,5,1,1,2,2,1,1,5,
%T A010285 13,1,1,1,1,565,4,2,1,3,1,1,2,1,1,2,1,2,1,1,1,1,6,1,2,3,
%U A010285 2,4,1,1,4,4,2,1,4,2,11,74,4,2,5,15,1,2,3,3,1,4,13,3,1
%N A010285 Continued fraction for cube root of 56.
%K A010285 nonn,cofr
%O A010285 0,1
%A A010285 njas
%I A030757
%S A030757 1,1,3,1,4,1,2,6,1,2,3,1,8,1,2,3,5,3,11,1,2,3,1,4,8,5,13,1,2,
%T A030757 4,4,3,6,10,7,16,1,2,3,1,5,1,6,4,9,12,9,18,1,2,3,1,4,2,2,6,2,
%U A030757 8,6,11,14,11,22,1,2,3,1,4,2,7,3,3,8,3,11,7,13,16,16,25,1,2,3
%N A030757 Row 1, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in rows 1 and 2, for j=m,m-1,...2,1, where m=max number in row 1 from stages 1 to k-1; stage 1 is 1 in row 1.
%K A030757 nonn
%O A030757 1,3
%A A030757 Clark Kimberling, ck6@cedar.evansville.edu
%I A004592
%S A004592 3,1,4,1,2,6,3,0,0,1,4,0,8,5,1,6,3,1,2,7,2,2,4,2,3,6,2,8,2,0,7,1,0,
%T A004592 6,7,0,4,0,0,1,6,7,1,0,7,6,1,6,4,3,5,3,3,5,2,4,6,2,6,7,2,2,0,6,6,8,
%U A004592 8
%N A004592 Expansion of sqrt(10) in base 9.
%K A004592 nonn,base,cons
%O A004592 1,1
%A A004592 njas
%I A010602
%S A010602 3,1,4,1,3,8,0,6,5,2,3,9,1,3,9,3,0,0,4,4,9,3,0,7,5,8,9,6,4,6,2,7,4,
%T A010602 9,9,2,6,3,5,0,8,5,9,7,1,8,5,0,0,7,2,6,4,2,5,6,4,2,3,5,2,6,1,7,3,3,
%U A010602 7,5,9,0,0,0,8,5,1,2,4,3,4,2,9,1,6,2,1,3,7,0,3,1,0,1,9,8,2,5,3,1,4
%N A010602 Decimal expansion of cube root of 31.
%K A010602 nonn,cons
%O A010602 1,1
%A A010602 njas
%I A029212
%S A029212 1,0,1,0,1,1,1,1,1,1,3,1,4,1,4,3,4,4,4,4,7,4,9,4,10,7,10,
%T A029212 9,10,10,14,10,17,10,19,14,20,17,20,19,25,20,29,20,32,25,
%U A029212 34,29,35,32,41,34,46,35,50,41,53,46,55,50,63,53,69,55
%N A029212 Expansion of 1/((1-x^2)(1-x^5)(1-x^10)(1-x^12)).
%K A029212 nonn
%O A029212 0,11
%A A029212 njas
%I A035687
%S A035687 0,0,0,0,0,0,1,0,1,0,1,1,1,1,3,1,4,1,4,3,4,4,7,4,10,4,11,8,11,11,15,
%T A035687 12,21,12,25,18,26,24,31,28,42,29,50,38,55,50,62,58,79,63,95,76,105,
%U A035687 96,118,113,144,123,172,145,193,178,213,208,255,230,302,262,340,316
%N A035687 Partitions into parts 8k+2 and 8k+5 with at least one part of each type
%K A035687 nonn,part
%O A035687 1,15
%A A035687 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A013705
%S A013705 3,1,4,1,5,9,0,6,5,3,5,8,9,7,9,3,2,4,0,4,6,2,6,4,3,3,8,3,2,6,9,5,0,
%T A013705 2,8,8,4,1,9,7,2,9,1,3,9,9,3,7,5,1,0,3,0,5,0,9,7,4,9,4,4,6,9,3,3,4,
%U A013705 9,8,1,6,4,0,0,8,8,0,6,8,0,5
%N A013705 Decimal expansion of 4 sum {k = 1...500,000} (-1)^(k-1)/(2k-1).
%D A013705 J. M. Borwein, P. R. Borwein and K. Dilcher, Pi, Euler numbers, and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
%H A013705 J. M. Borwein and R. M. Corless, Review of ``An Encyclopedia of Integer Sequences'' by N. J. A. Sloane and Simon Plouffe", SIAM Review, 38 (June 1996), 333-337.
%e A013705 3.1415906535897932404626433832695028841972913993751030509749446933498164008806805...
%Y A013705 Cf. A000796, A013707, A013706.
%K A013705 cons,nonn
%O A013705 1,1
%A A013705 njas
%I A000796 M2218 N0880
%S A000796 3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,
%T A000796 2,8,8,4,1,9,7,1,6,9,3,9,9,3,7,5,1,0,5,8,2,0,9,7,4,9,4,4,5,9,2,3,0,
%U A000796 7,8,1,6,4,0,6,2,8,6,2,0,8,9,9,8,6,2,8,0,3,4,8,2,5,3,4,2,1,1,7,0,6
%N A000796 Decimal expansion of Pi.
%D A000796 D. Shanks and J. W. Wrench. Jr., Calculation of pi to 100,000 decimals. Math. Comp. 16 1962 76-99.
%H A000796 Daniel Sedory, The Pi Pages
%H A000796 UCLA Repository, Pi to 100,000 digits, but incorrect after the 15,094-th digit
%H A000796 E. W. Weisstein, Link to a section of The World of Mathematics.
%e A000796 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303820...
%K A000796 cons,nonn,nice
%O A000796 1,1
%A A000796 njas
%E A000796 Warning: There are some erroneous files on the Net that claim to give large numbers of digits of the decimal expansion of Pi. The error usually occurs after the 15094-th digit.
%E A000796 Additional comments from William Rex Marshall )w.r.marshall@actrix.co.nz), Apr 20, 2001
%I A057466
%S A057466 3,1,4,1,5,9,8,2,8,0,6,5,8,3,2,0,1,3,6,1,7,1,2,3,4,6,7,5,1,3,3,7,5,2,0,
%T A057466 6,6,7,2,1,9,9,3,6,1,8,9,7,5,2,9,8,5,2,7,4,7,5,4,1,1,6,7,6,7,8,1,2,2,8,
%U A057466 0,5,8,3,8,6,7,8,8,5,3,0,1,5,6,0,5,4,2,9,6,4,4,3,6,7,6,0,3,6
%N A057466 The ninth root of 10*e^8.
%C A057466 "There are many approximate relations between e and [Pi], most of them rather crude. This one is surprisingly accurate: ... = 3.14159 828 ... [.] [Michele Fanelli]"
%D A057466 David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 28.
%t A057466 RealDigits[N[(10*E^8)^(1/9), 100]][[1]]
%K A057466 nonn
%O A057466 0,1
%A A057466 Robert G. Wilson v (rgwv@kspaint.com), Dec 07 2000
%I A014462
%S A014462 1,1,1,3,1,4,1,5,10,1,6,15,1,7,21,35,1,8,28,56,1,9,36,84,126,1,10,45,
%T A014462 120,210,1,11,55,165,330,462,1,12,66,220,495,792,1,13,78,286,715,1287,
%U A014462 1716,1,14,91,364,1001,2002,3003,1,15,105,455,1365,3003,5005,6435,1,16
%N A014462 Triangular array formed from elements to left of middle of Pascal's triangle.
%Y A014462 Cf. A014413.
%K A014462 tabf,nonn,easy
%O A014462 1,4
%A A014462 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A014462 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A016474
%S A016474 3,1,4,1,5,11,2,57,1,350,1,5,4,6,1,8,1,1,2,2,2,3,3,2,9,
%T A016474 1,3,3,1,3,2,25,1,2,2,24,8,1,1,1,11,1,3,6,1,13,1,3,1,1,
%U A016474 1,1,1,21,2,1,1,2,2,1,2,7,11,3,1,1,13,5,1,5,4,1,4,1,5,2
%N A016474 Continued fraction for ln(46).
%K A016474 nonn,cofr
%O A016474 1,1
%A A016474 njas
%I A064575
%S A064575 1,1,3,1,4,1,6,3,6,1,11,1,8,4,14,1,16,1,19,5,16,1,29,3,22,7,31,1,37,1,
%T A064575 42,7,38,4,62,1,48,9,69,1,73,1,80,14,76,1,114,3,100,11,121,1,132,5,150,
%U A064575 14,142,1,193,1,168,20,213,5,223,1,247,17,247,1,319,1,286,25,339,4,355
%N A064575 First differences of A064572.
%C A064575 Apparently a(n)=1 when n+1 is prime.
%Y A064575 A064572, A064573, A064574, A064576, A064577, A028422.
%K A064575 easy,nonn
%O A064575 1,3
%A A064575 Marc Le Brun (mlb@well.com), Sep 20 2001
%I A006022 M2219
%S A006022 0,1,1,3,1,4,1,7,4,6,1,10,1,8,6,15,1,13
%N A006022 Maundy cake values.
%D A006022 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 28.
%K A006022 nonn
%O A006022 1,4
%A A006022 njas
%I A014413
%S A014413 1,1,3,1,4,1,10,5,1,15,6,1,35,21,7,1,56,28,8,1,126,84,36,9,1,210,120,
%T A014413 45,10,1,462,330,165,55,11,1,792,495,220,66,12,1,1716,1287,715,286,78,
%U A014413 13,1,3003,2002,1001,364,91,14,1,6435,5005,3003,1365,455,105,15,1
%N A014413 Triangular array formed from elements to right of middle of Pascal's triangle.
%Y A014413 Cf. A014462.
%K A014413 tabf,nonn,easy
%O A014413 1,3
%A A014413 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A014413 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A051348
%S A051348 1,1,1,3,1,4,1,21,17,11,1,72,1,29,61,987,1,1292,1,1353,421,199,1,23184,
%T A051348 15005,521,98209,24447,1,83204,1,2178309,19801,3571,141961,7465176,1,
%U A051348 9349,135721,20466831,1,10304396,1,7880997,113490317,64079,1
%N A051348 F_n/product{p|n}[ F_p ], where F_k is k_th Fibonacci number and the p's in product are the distinct primes dividing n.
%e A051348 a[ 15 ]=F_15/(F_3*F_5)=610/(2*5)=61
%K A051348 nonn
%O A051348 1,4
%A A051348 Leroy Quet (qqquet@mindspring.com)
%I A020851
%S A020851 1,0,3,1,4,2,1,2,4,6,2,5,8,7,9,3,4,0,7,2,4,9,8,8,0,9,7,3,4,9,4,1,8,
%T A020851 2,2,7,5,9,2,0,4,4,1,5,9,4,4,6,3,2,6,3,7,8,6,2,2,7,6,2,9,5,1,9,8,1,
%U A020851 2,0,2,5,5,1,7,8,2,6,9,4,6,8,8,6,0,0,8,8,5,7,3,0,6,2,2,0,9,4,8,6,6
%N A020851 Decimal expansion of 1/sqrt(94).
%K A020851 nonn,cons
%O A020851 0,3
%A A020851 njas
%I A048225
%S A048225 3,1,4,2,3,6,5,7,8,11,9,14,16,17,20,10,19,24,26,27,30,12,22,31,36,38,
%T A048225 39,42,13,25,35,44,49,51,52,55,15,28,40,50,59,64,66,67,70,18,33,46,58,
%U A048225 68,77,82,84,85,88,21,39,54,67,79,89,98,103,105
%N A048225 Triangular array T by rows: T(i,j)=b(i+1)-b(i+1-j); j=1,2,...,i; i=1,2,3,...; b=A048224.
%e A048225 Rows: {3}; {1,4}; {2,3,6}; ...
%K A048225 nonn,tabl
%O A048225 1,1
%A A048225 Clark Kimberling, ck6@cedar.evansville.edu
%I A065256
%S A065256 0,3,1,4,2,15,18,16,19,17,5,8,6,9,7,20,23,21,24,22,10,13,11,14,12,75,
%T A065256 78,76,79,77,90,93,91,94,92,80,83,81,84,82,95,98,96,99,97,85,88,86,89,
%U A065256 87,25,28,26,29,27,40,43,41,44,42,30,33,31,34,32,45,48,46,49,47,35,38
%N A065256 Quintal Queens permutation of N: halve or multiply by 3 (mod 5) each digit (0->0, 1->3, 2->1, 3->4, 4->2) of the base-5 representation of n.
%C A065256 All the permutations A004515, and A065256-A065258 consist of the first fixed term ("Queen on the corner") plus infinitely many 4-cycles, and they satisfy the "non-attacking queen condition" that p(i+d) <> p(i)+-d for all i and d >= 1.
%C A065256 The corresponding infinite permutation matrix is a scale-invariant fractal (Cf. A048647), and any subarray (5^i)x(5^i) (i >= 1) cut from its corner gives a solution to the case n=5^i of the n nonattacking queens on nxn chess-board (A000170). Is there any permutation of N which would give solutions to the queen problem with more frequent intervals than A000351 ?
%H A065256 Index entries for sequences that are permutations of the natural numbers
%p A065256 [seq(QuintalQueens0Inv(j),j=0..124)];
%p A065256 HalveDigit := (d,b) -> op(2,op(1,msolve(2*x=d,b))); # b should be an odd integer >= 3, and d should be in range [0,b-1].
%p A065256 HalveDigits := proc(n,b) local i; add((b^i)*HalveDigit((floor(n/(b^i)) mod b),b),i=0..floor(evalf(log[b](n+1)))+1); end;
%p A065256 QuintalQueens0Inv := n -> HalveDigits(n,5);
%Y A065256 Inverse permutation: A004515. A065256[n] = A065258[n+1]-1. Cf. also A065187, A065189.
%K A065256 nonn
%O A065256 0,2
%A A065256 Antti.Karttunen@iki.fi Oct 26 2001
%I A016573
%S A016573 3,1,4,2,17,6,1,2,1,4,34,1,1,1,2,5,1,4,2,1,1,2,30,2,7,4,
%T A016573 5,2,1,2,1,2,3,1,1,1,9,6,2,5,1,7,2,1,1,1,2,1,1,4,1,3,1,
%U A016573 13,1,5,1,7,3,1,1,1,2,37,3,1,20,1,11,1,7,1,2,1,1,18,4,1
%N A016573 Continued fraction for ln(91/2).
%K A016573 nonn,cofr
%O A016573 1,1
%A A016573 njas
%I A055171
%S A055171 1,1,1,1,3,1,4,3,1,1,6,3,2,4,1,1,8,3,3,4,2,6,1,2,1,1,11,3,5,4,3,6,2,2,
%T A055171 3,8,1,1,13,3,8,4,4,6,3,2,5,8,2,11,1,5,1,1,16,3,10,4,6,6,4,2,7,8,4,11,
%U A055171 2,5,3,13,1,1,18,3,12,4,9,6,6,2,9,8,5,11,3,5
%N A055171 Cumulative counting sequence: method B (noun,adjective)-pairs with 1st term 1.
%C A055171 Segments (as in %e): 1; 1,1; 1,3; 1,4,3,1; 1,6,3,2,4,1; ...
%C A055171 Conjecture: every positive integer occurs.
%e A055171 Write 1, thus having 1 1 time, thus having 1 3 times, thus having 1 4 times and 3 1 time, etc.
%K A055171 nonn
%O A055171 1,5
%A A055171 Clark Kimberling, ck6@cedar.evansville.edu, Apr 27 2000
%I A064883
%S A064883 1,3,1,4,3,1,5,4,7,3,1,6,5,9,4,11,7,10,3,1,7,6,11,5,14,9,13,4,15,11,18,
%T A064883 7,17,10,13,3,1,8,7,13,6,17,11,16,5,19,14,23,9,22,13,17,4,19,15,26,11,
%U A064883 29,18,25,7,24,17,27,10
%N A064883 Eisenstein array Ei(1,3).
%C A064883 In Eisenstein's notation this is the array for m=1 and n=3; see pp.41-2 of the Eisenstein ref. given for A064881. The array for m=n=1 is A049456.
%C A064883 For n >= 1, the number of entries of row n is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 4*A007051(n-1).
%C A064883 The binary tree built from the rationals a(n,m)/a(n,m+1), m=0..2^(n-1), for each row n >= 1 gives the sub-tree of the (Eisenstein-)Stern-Brocot tree in the version of, e.g., Calkin and Wilf (for the ref. see A002487, also for the Wilf link) with root 1/3. The composition rule for this tree is i/j -> i/(i+j), (i+j)/j.
%F A064883 a(n,m)= a(n-1,m/2) if m is even, else a(n,m)= a(n-1,(m-1)/2)+a(n-1,(m+1)/2, a(1,0)=1, a(1,1)=3.
%e A064883 {1,3}; {1,4,3}; {1,5,4,7,3}; {1,6,5,9,4,11,7,10,3}; ...
%e A064883 This binary subtree of rationals is built from 1/3; 1/4,4/3; 1/5,5/4,4/7,7/3; ..
%K A064883 nonn,easy,tabf
%O A064883 1,2
%A A064883 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Oct 19 2001
%I A008314
%S A008314 1,1,1,1,1,3,1,4,3,1,5,10,1,6,15,10,1,7,21,35,1,8,28,56,35,1,9,36,
%T A008314 84,126,1,10,45,120,210,126,1,11,55,165,330,462,1,12,66,220,
%U A008314 495,792,462
%N A008314 Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x).
%D A008314 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
%H A008314 Index entries for sequences related to Chebyshev polynomials.
%K A008314 nonn,tabl
%O A008314 0,6
%A A008314 njas
%I A030758
%S A030758 1,1,3,1,4,3,1,6,4,3,2,1,8,6,4,3,2,1,11,8,6,5,4,3,2,1,13,11,8,
%T A030758 6,5,4,3,2,1,16,13,11,10,8,7,6,5,4,3,2,1,18,16,13,12,11,10,9,
%U A030758 8,7,6,5,4,3,2,1,22,18,16,14,13,12,11,10,9,8,7,6,5,4,3,2,1,25
%N A030758 Row 2, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in rows 1 and 2, for j=m,m-1,...2,1, where m=max number in row 1 from stages 1 to k-1; stage 1 is 1 in row 1.
%K A030758 nonn
%O A030758 1,3
%A A030758 Clark Kimberling, ck6@cedar.evansville.edu
%I A029151
%S A029151 1,0,1,1,1,1,3,1,4,3,4,4,7,4,9,7,10,9,14,10,17,14,19,17,
%T A029151 25,19,29,25,32,29,40,32,46,40,50,46,60,50,68,60,74,68,
%U A029151 86,74,96,86,104,96,119,104,131,119,141,131,159,141,174
%N A029151 Expansion of 1/((1-x^2)(1-x^3)(1-x^6)(1-x^8)).
%K A029151 nonn
%O A029151 0,7
%A A029151 njas
%I A054019
%S A054019 3,1,4,3,5,4,5,7,4,1,8,11,6,9,8,13,8,6,10,15,8,6,11,6,13,7,19,15,1,12,
%T A054019 21,21,12,16,20,14,8,21,1,12,19,8,20,21,27,8,14,12,27,10,29,27,5,20,16,
%U A054019 35,10,27,35,31,30,29,3,12,28,5,1,35,26,10,20,37,12,33,18,43,43,45,22
%N A054019 Square roots of A054018.
%Y A054019 Cf. A048050, A054013, A054017, A054018.
%K A054019 nonn
%O A054019 1,1
%A A054019 Asher Auel (asher.auel@reed.edu) Jan 19, 2000
%I A035626
%S A035626 0,0,0,0,1,0,1,1,3,1,4,3,7,4,10,8,15,11,21,18,30,24,42,37,56,50,78,70,
%T A035626 102,95,137,129,179,171,236,227,303,297,395,386,502,501,643,641,814,
%U A035626 820,1030,1041,1291,1317,1622,1652,2018,2075,2509,2582,3107,3212,3834
%N A035626 Partitions into parts 4k+2 and 4k+3 with at least one part of each type
%K A035626 nonn,part
%O A035626 1,9
%A A035626 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A060043
%S A060043 1,3,1,4,3,7,9,6,1,15,12,3,20,8,9,45,15,22,67,13,1,42,99,18,3,84,
%T A060043 135,12,9,140,175,28,22,231,231,14,51,351,306,24,1,97,551,354,24,3,
%U A060043 188,783,465,31
%N A060043 Triangle T(n,k), n >= 1, k >= 1, of generalized sum of divisors function, read by rows.
%C A060043 Lengths of rows are 1 1 2 2 2 3 3 3 3 ... (A003056).
%D A060043 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
%F A060043 T(n,1) = sum of divisors of n (A000203), T(n,k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*m_1 + s_2*m_2 + ... + s_k*m_k = n and the sum is over all such k-partitions of n.
%F A060043 G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1-q^m_1)*(1-q^m_2)*...*(1-q^m_k))^2 = Sum_n T(n,k)*q^n.
%e A060043 Triangle turned on its side begins:
%e A060043 1 3 4 7 6 12 .8 15 13 18 etc
%e A060043 . . 1 3 9 15 30 45 67 99 etc
%e A060043 . . . . . .1 .3 .9 22 42 etc
%e A060043 . . . . . . . . .. .. .1 etc
%e A060043 For example, T(6,2) = 15.
%Y A060043 Diagonals give A000203, A002127, A002128. Cf. A060044.
%K A060043 nonn,tabf,easy,nice,more
%O A060043 1,2
%A A060043 njas, Mar 19 2001
%I A054907
%S A054907 1,0,0,0,0,0,0,0,1,0,0,0,1,0,1,1,3,1,4,3,12,12,28,49,180,368
%N A054907 Number of n-dimensional unimodular lattice (or quadratic forms) containing no vectors of norm 1.
%D A054907 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
%Y A054907 Cf. A005134, A054908-A054909, A054911.
%K A054907 nonn,nice,hard
%O A054907 0,17
%A A054907 njas, May 23 2000
%I A014388
%S A014388 3,1,4,4,5,3,6,11,7,15,8,13,9,17,10,24,11,23,12,73,13,3000,
%T A014388 14,11000,15,15000,16,101,17,104,18,103,19,111,20,115
%N A014388 a(2n-1) = n+2, a(2n) = smallest number requiring n+2 letters in English.
%Y A014388 Cf. A000916, A001166, A014388, A045494, A045495.
%K A014388 nonn,word,easy
%O A014388 1,1
%A A014388 jhau@win.tue.nl (Jacques Haubrich)
%I A021322
%S A021322 0,0,3,1,4,4,6,5,4,0,8,8,0,5,0,3,1,4,4,6,5,4,0,8,8,0,5,0,3,1,4,4,6,
%T A021322 5,4,0,8,8,0,5,0,3,1,4,4,6,5,4,0,8,8,0,5,0,3,1,4,4,6,5,4,0,8,8,0,5,
%U A021322 0,3,1,4,4,6,5,4,0,8,8,0,5,0,3,1,4,4,6,5,4,0,8,8,0,5,0,3,1,4,4,6,5
%N A021322 Decimal expansion of 1/318.
%K A021322 nonn,cons
%O A021322 0,3
%A A021322 njas
%I A036412
%S A036412 0,0,0,0,1,0,0,1,1,3,1,4,4,7,5,5,6,4,4,6,7,6,8,5,2,6,4,5,3,4,3,0,3,2,
%T A036412 0,3,3,3,0,4,4,5,6,5,6,7,8,8,8,9,8,8,7,8,8,8,9,8,8,7,6,7,6,1,5,4,4,3,
%U A036412 2,2,0,5,4,3,5,5,6,2,8,9,9,10,11,9,11,13,16,14,16,16,17,17,18,18,20
%N A036412 Number of empty intervals when fractional_part(i E) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.
%H A036412 E. W. Weisstein, Link to a section of The World of Mathematics.
%H A036412 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A036412 Cf. A036413.
%K A036412 nonn
%O A036412 1,10
%A A036412 Eric W. Weisstein (eric@weisstein.com)
%I A016473
%S A016473 3,1,4,5,1,4,10,2,3,3,1,1,3,1,1,3,6,5,2,8,1,1,3,15,1,5,
%T A016473 3,3,10,1,31,1,5,1,2,7,5,2,1,1,5,1,1,1,1,1,44,1,1,2,1,1,
%U A016473 2,3,6,1,18,5,10,1,5,1,26,1,10,1,12,1,1,1,4,2,1,3,2,1,1
%N A016473 Continued fraction for ln(45).
%K A016473 nonn,cofr
%O A016473 1,1
%A A016473 njas
%I A029637
%S A029637 1,1,1,3,1,4,5,1,5,9,7,1,6,14,16,9,1,7,20,30,25,11,1,8,27,50,55,36,13,1,
%T A029637 9,35,77,105,91,49,15,1,10,44,112,182,196,140,64,17,1,11,54,156,294,378,
%U A029637 336,204,81,19,1,12,65,210,450,672,714,540,285,100,21,1,13,77,275,660
%N A029637 Numbers in the (1,2)-Pascal triangle A029635 that are different from 2.
%K A029637 nonn,tabf
%O A029637 0,4
%A A029637 Mohammad K. Azarian, ma3@cedar.evansville.edu
%E A029637 More terms from dww
%I A046070
%S A046070 3,1,4,5,3,26,7,2,4,3,2,6,9,2,16,5,3,6,2553,24,10,31,2,14,5,9,6,3,2,
%T A046070 16,5,3,6,9,4,14,11,3,4,3,5,4,11,2,8,3,4,6,9,4,18,7,3,12,149,3,14,3,2,
%U A046070 16,3,3,4,113,3,14,11,9,18,5,2,4,13,2,16,221,4,8,5,4,6,31,3,6,5,3,4,3
%N A046070 Second smallest m such that (2n-1)2^m-1 is prime, or -1 if no such value exists.
%C A046070 There exist odd integers 2k-1 such that (2k-1)2^n-1 is always composite.
%D A046070 Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 357-359, 1996.
%H A046070 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A046070 Cf. A046068, A046069.
%K A046070 nonn
%O A046070 1,1
%A A046070 Eric W. Weisstein (eric@weisstein.com)
%I A050057
%S A050057 1,3,1,4,5,9,10,13,14,27,37,46,51,55,56,59,60,119,175,230,281,327,364,
%T A050057 391,405,418,428,437,442,446,447,450,451,901,1348,1794,2236,2673,3101,
%U A050057 3519,3924,4315,4679,5006,5287,5517,5692,5811
%N A050057 a(n)=a(n-1)+a(m), where m=2^(p+1)+2-n, and 2^p= 4.
%K A050057 nonn
%O A050057 1,2
%A A050057 Clark Kimberling, ck6@cedar.evansville.edu
%I A051203
%S A051203 3,1,4,5,35,10,8,26,15,38,20,13,55,78,27,70,68,53,36,282,44,73,75,69,
%T A051203 64,34,32,585,51,30,139,165,72,121,535,97,83,253,67,469,168,61,147,146,
%U A051203 59,93,123,286,815,1398,112,294,119,129,347,138,124,81,144,194,256,142
%N A051203 Least inverse of A005210.
%K A051203 nonn
%O A051203 0,1
%A A051203 dww
%I A060922
%S A060922 1,3,1,4,6,1,7,17,9,1,11,38,39,12,1,18,80,120,70,15,1,29,158,315,280,
%T A060922 110,18,1,47,303,753,905,545,159,21,1,76,566,1687,2568,2120,942,217,24,
%U A060922 1,123,1039,3612,6666,7043,4311
%N A060922 Convolution triangle for Lucas numbers A000032(n+1), n >= 0.
%C A060922 In the language of Shapiro et al. (see A053121 for the ref.) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. G.f. for row polynomials p(n,x):= sum(a(n,m)*x^m,m=0..n) is (1+2*z)/(1-(1+x)*z-(1+2*x)*z^2).
%C A060922 Row sums give A060925. Column sequences (without leading zeros) are, for m=0..6: A000032(n+1)= A000204(n+1) (Lucas), A004799(n+1), A060929-33.
%C A060922 Bisection of this triangle gives triangles A060923 (even part) and A060924 (odd part).
%C A060922 For the m-th column sequence (without leading zeros) one has: a(n+m,m)= (pL1(m,n)*L(n+2)+pL2(m,n)*L(n+1))/(m!*5^m), m >= 0, with the Lucas numbers L(n)=A000032(n), n >= 0, and the row polynomials pL1(n,x):=sum(A061188(n,m)*x^n,m=0..n) and pL2(n,x):= sum(A061189(n,m)*x^m,m=0..n).
%F A060922 a(n,m)=((n-m+1)*a(n,m-1)+2*(2*n-m)*a(n-1,m-1)+4*(n-1)*a(n-2,m-1))/(5*m), n >= m >= 1, a(n,0)= A000204(n+1)= A000032(n+1).
%F A060922 G.f. for m-th column: ((1+2*x)/(1-x-x^2))* ((x*(1+2*x))/(1-x-x^2))^m.
%e A060922 {1}; {3,1}; {4,6,1}; ...; p(2,x) = 4+6*x+x^2.
%K A060922 nonn,easy,tabl
%O A060922 0,2
%A A060922 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Apr 20 2001
%I A011089
%S A011089 3,1,4,6,3,4,6,2,8,3,6,4,5,7,8,8,6,2,0,6,2,1,8,9,2,6,4,2,2,8,2,8,1,
%T A011089 3,8,1,5,6,1,8,5,6,0,2,3,8,0,6,6,2,4,4,6,2,4,0,2,2,3,9,2,8,9,0,8,2,
%U A011089 0,3,3,7,3,9,6,0,5,3,5,4,9,5,3,5,5,4,2,0,0,5,6,3,1,0,5,6,2,1,9,7,2
%N A011089 Decimal expansion of 4th root of 98.
%K A011089 nonn,cons
%O A011089 1,1
%A A011089 njas
%I A024934
%S A024934 0,1,1,3,1,4,6,7,4,8,8,13,10,8,12,18,20,27,28,26,21,29,33,37,31,37,37,46,46,
%T A024934 56,65,62,54,53,59,70,61,57,62,74,75,88,89,95,84,98,108,116,124,119,119,134,
%U A024934 145,145,152,146,131,147,154,171,156,164,180,180,182,200,200,193,198,217
%N A024934 a(n) = sum of remainders of n mod p(k), over all k such that p(k) < n.
%K A024934 nonn
%O A024934 2,4
%A A024934 Clark Kimberling (ck6@cedar.evansville.edu)
%I A049918
%S A049918 1,3,1,4,6,14,26,54,105,213,424,850,1697,3392,6776,13540,27052,54157,
%T A049918 108312,216626,433249,866496,1732984,3465956,6931884,13863717,27727326,
%U A049918 55454441,110908456,221816065,443630435,887257486,1774508208
%N A049918 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=n-1-2^p, and 2^p= 4.
%K A050059 nonn
%O A050059 1,2
%A A050059 Clark Kimberling, ck6@cedar.evansville.edu
%I A025121
%S A025121 1,1,3,1,4,8,14,14,21,19,27,25,34,33,43,42,54,68,83,81,97,95,112,111,130,
%T A025121 129,149,147,170,168,191,190,215,213,238,235,262,290,321,319,351,349,382,
%U A025121 381,416,414,451,449,488,486,526,525,566,565,607,603,647,643,687,685,731
%N A025121 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = A000201 (lower Wythoff sequence), t = A023534.
%K A025121 nonn
%O A025121 1,3
%A A025121 Clark Kimberling (ck6@cedar.evansville.edu)
%I A025097
%S A025097 1,1,3,1,4,8,15,15,25,23,40,37,62,55,91,73,120,196,318,316,514,511,829,822,
%T A025097 1332,1314,2128,2081,3369,3246,5254,4932,7982,7138,11550,9342,15117,24462,
%U A025097 39582,39571,64029,64000,103556,103480,167436,167237,270597,270076,436994
%N A025097 s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (Lucas numbers), t = A023534.
%K A025097 nonn
%O A025097 1,3
%A A025097 Clark Kimberling (ck6@cedar.evansville.edu)
%I A049916
%S A049916 1,3,1,4,8,16,32,57,90,211,422,837,1650,3242,6152,11076,17650,41451,
%T A049916 82902,165797,331570,663082,1325832,2650436,5296370,10581242,21097232,
%U A049916 41945796,82897330,161824122,307847382,553894666,882839280
%N A049916 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=2n-3-2^(p+1), and 2^pIndex entries for sequences that are permutations of the natural numbers
%Y A065253 Cf. A064823 A000796 A065254.
%K A065253 base,nice,nonn
%O A065253 1,1
%A A065253 Klaus Strassburger (strass@ddfi.uni-duesseldorf.de), Oct 26 2001
%I A010756
%S A010756 0,0,1,0,1,3,1,4,11,5,16,41,22,63,155,92,247,591,376,967,2267,1518,3785,
%T A010756 8735,6085,14820,33775,24285,58060,130965,96647,227612,509015,383911,
%U A010756 892926,1982269,1523117,3505386,7732659,6037745,13770404,30208749
%N A010756 Sum along upward diagonal of Pascal triangle from (but not including) center.
%K A010756 nonn
%O A010756 0,6
%A A010756 rkg@cpsc.ucalgary.ca (Richard Guy)
%I A010284
%S A010284 3,1,4,13,2,1,6,1,8,1,5,12,15,1,5,3,1,3,3,8,9,2,1,3,1,6,
%T A010284 1,1,2,4,8,2,1,1,6,1,4,1,1,17,6,4,1,19,1,3,1,1,1,6,3,1,
%U A010284 1,1,2,1,1,1,4,1,2,1,2,1,1,7,5,7,1,1,67,11,1,1,7,2,2,2
%N A010284 Continued fraction for cube root of 55.
%K A010284 nonn,cofr
%O A010284 0,1
%A A010284 njas
%I A064809
%S A064809 3,1,4,15,9,2,6,5,35,8,97,93,23,84,62,64,33,83,27,950,28,841,971,69,39,
%T A064809 937,510,58,20,974,94,45,92,30,7,81,640,628,620,89,98,6280,34,82,53,42,
%U A064809 11,70,67,982,14,80,86,51,32,8230,66,470,938,44,60,95,50,582,231,72
%N A064809 Decimal digits of Pi written as a sequence of natural numbers avoiding duplicates.
%C A064809 Start with the first digit of Pi and set a(1)=3. Let p(1),...,p(i), be the digits of Pi used to construct a(1),...,a(n); then a(n+1) is the smallest integer with digits p(i+1),..,p(i+j) such that a(n+1) is new and p(i+j+1) not= 0.
%C A064809 Is the sequence is a permutation of the natural numbers?
%e A064809 Pi=3.141592653589...
%Y A064809 Cf. A000796.
%K A064809 base,nonn
%O A064809 1,1
%A A064809 Klaus Strassburger (strass@ddfi.uni-duesseldorf.de), Oct 22 2001
%I A058361
%S A058361 3,1,4,15,22,121,735,31,46,22143,4468,67,31455,391,2308,447,94,33151,
%T A058361 16383,139,202,7551,5224,787,1595391,3685,580,30591,418,42495,1791,607,
%U A058361 1342,3217407,1095166,283,398847,32767,365311,88575,1174,6925,12304383
%N A058361 a(n) is the least k in A002977 with a gap of n. Also n + a(n) is the least k in A007448 which is repeated n times.
%t A058361 k = {1}; Do[ k = Union[ Join[ k, 2k + 1, 3k + 1 ] ]; l = Length[ k ]; i = 1; While[ i < l && k[ [ i ] ] < 10^9, i++ ]; k = Take[ k, {1, i} ], {n, 1, 30} ]; f[ n_Integer ] := (i = 1; While[ k[ [ i + 1 ] ] - k[ [ i ] ] != n, i++ ]; k[ [ i ] ]); Table[ f[ n ], {n, 1, 84} ]
%Y A058361 Cf. A002977, A007448.
%K A058361 nonn
%O A058361 1,1
%A A058361 Robert G. Wilson v (rgwv@kspaint.com), Dec 16 2000
%I A016734
%S A016734 1,1,3,1,4,18,2,330,3,1,2,1,1,4,1,4,2,6,2,2,1,3,4,5,1,6,
%T A016734 3,1,5,1,37,1,1,1,3,1,2,1,1,2,1,1,5,4,2,2,1,37,4,31,1,1,
%U A016734 49,1,7,1,6,2,7,2,2,4,2,6,1,1,8,1,1,2,9,1,5,1,12,1,10,2
%N A016734 Continued fraction for ln(6).
%K A016734 nonn,cofr
%O A016734 1,3
%A A016734 njas
%I A010286
%S A010286 3,1,5,1,1,1,1,55,1,3,1,2,1,13,1,3,1,1,2,4,1,1,3,2,1,2,
%T A010286 1,1,1,4,2,1,6,1,1,6,1,5,1,1,1,2,4,27,7,1,9,1,1,2,2,2,2,
%U A010286 1,2,4,10,1,1,9,9,1,2,3,1,1,23,6,8,18,1,2,1,1,1,1,24,15
%N A010286 Continued fraction for cube root of 57.
%K A010286 nonn,cofr
%O A010286 0,1
%A A010286 njas
%I A002945 M2220
%S A002945 1,3,1,5,1,1,4,1,1,8,1,14,1,10,2,1,4,12,2,3,2,1,3,4,1,1,2,14,3,12,1,
%T A002945 15,3,1,4,534,1,1,5,1,1,121,1,2,2,4,10,3,2,2,41,1,1,1,3,7,2,2,9,4,1,3,7,
%U A002945 6
%N A002945 Continued fraction for cube root of 2.
%D A002945 S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
%H A002945 Index entries for continued fractions for constants
%K A002945 cofr,nonn
%O A002945 1,2
%A A002945 njas
%I A016475
%S A016475 3,1,5,1,2,16,1,1,2,2,1,3,1,5,32,1,10,1,1,1,3,1,19,3,4,
%T A016475 1,19,1,6,1,3,2,2,2,2,250,2,1,1,5,3,1,3,4,23,7,2,3,19,3,
%U A016475 1,1,43,1,10,4,5,1,1,129,1,1,1,1,3,2,3,2,1,3,9,94,2,6,3
%N A016475 Continued fraction for ln(47).
%K A016475 nonn,cofr
%O A016475 1,1
%A A016475 njas
%I A037227
%S A037227 1,3,1,5,1,3,1,7,1,3,1,5,1,3,1,9,1,3,1,5,1,3,1,7,1,3,1,5,1,3,1,11,1,3,
%T A037227 1,5,1,3,1,7,1,3,1,5,1,3,1,9,1,3,1,5,1,3,1,7,1,3,1,5,1,3,1,13,1,3,1,5,
%U A037227 1,3,1,7,1,3,1,5,1,3,1,9,1,3,1,5,1,3,1,7,1,3,1,5,1,3,1,11,1,3,1,5,1,3
%N A037227 If n = 2^m*k, k odd, then a(n)=2*m+1.
%D A037227 D. B. Shapiro, Problem 10456, Amer. Math. Monthly, 105 (1998), 565-566.
%K A037227 nonn,easy,nice
%O A037227 1,2
%A A037227 njas
%E A037227 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A056753
%S A056753 1,3,1,5,1,3,1,7,1,3,1,9,1,3,1,7,1,3,1,11,1,3,1,7,1,3,1,13,1,3,1,7,1,3,
%T A056753 1,15,1,3,1,7,1,3,1,17,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,19,1,3,1,7,1,3,1,
%U A056753 15,1,3,1,7,1,3,1,21,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,23,1,3,1,7,1,3,1
%N A056753 Only odd numbers occur, and for all k there are k numbers between any two successive occurrences of k.
%C A056753 Only the numbers 2^m - 1 occur more than once.
%K A056753 nice,nonn
%O A056753 0,2
%A A056753 Claude Lenormand (hlne.lenormand@voonoo.net), Jan 19 2001
%I A001051
%S A001051 1,3,1,5,1,5,1,7,1,5,1,8,1,5,1,7,1,5,1,7,1,5,1,10,1,5,1,7,1,5,1,7,1,5,1,7,
%T A001051 1,5,1,7,1,5,1,7,1,5,1,8,1,5,1,7,1,5,1,7,1,5,1,8,1,5,1,7,1,5,1,7,1,5,1,7,
%U A001051 1,5,1,7,1,5,1,7,1,5,1,7,1,5,1,7,1,5,1,7,1,5,1,7
%N A001051 Subgroups of order n in orthogonal group O(3).
%H A001051 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A001051 Has period 1 5 1 7 except that a(2) = 3, a(4) = 5, a(12) = 8, a(24) = 10, a(48) = a(60) = a(120) = 8.
%Y A001051 The main sequences concerned with group theory are A000001, A000679, A001034, A001051, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A051881.
%K A001051 nonn,easy,nice
%O A001051 1,2
%A A001051 J. H. Conway, conway@math.princeton.edu
%I A002972 M2221
%S A002972 1,3,1,5,1,5,7,5,3,5,9,1,3,7,11,7,11,13,9,7,1,15,13,15,1,13,9,5,17,
%T A002972 13,11,9,5,17,7,17,19,1,3,15,17,7,21,19,5,11,21,19,13,1,23,5,17,19,25,13
%N A002972 From quadratic partitions of primes.
%D A002972 E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 243.
%K A002972 nonn
%O A002972 5,2
%A A002972 njas
%I A046730
%S A046730 1,3,1,5,1,5,7,5,3,5,9,1,3,7,11,7,11,13,9,7,1,15,13,15,1,13,9,5,
%T A046730 17,13,11,9,5,17,7,17,19,1,3,15,17,7,21,19,5,11,21,19,13,1,23,5,
%U A046730 17,19,25,13,25,23,1,5,15,27,9,19,25,17,11,5,25,27,23,29,29,25
%V A046730 -1,3,1,-5,-1,5,7,-5,-3,5,9,-1,3,-7,-11,7,11,-13,-9,-7,-1,15,13,-15,1,-13,-9,5,
%W A046730 -17,13,11,9,-5,17,7,-17,19,1,-3,15,17,-7,21,19,-5,-11,-21,19,13,1,-23,5,
%X A046730 -17,-19,25,-13,-25,-23,-1,-5,15,27,-9,-19,25,-17,11,5,-25,27,23,29,-29,25
%N A046730 A002172/2.
%K A046730 sign,done
%O A046730 0,2
%A A046730 njas
%I A029652
%S A029652 1,1,1,3,1,5,1,5,9,7,1,9,1,7,25,11,1,27,55,13,1,9,35,77,105,91,49,15,1,
%T A029652 17,1,11,81,19,1,65,285,21,1,13,77,275,825,385,121,23,1,935,2079,25,1,15,
%U A029652 1287,2717,4719,3289,169,27,1,119,1729,7007,9867,5005,819,29,1,17,135
%N A029652 Odd numbers in the (1,2)-Pascal triangle A029635.
%K A029652 nonn
%O A029652 0,4
%A A029652 Mohammad K. Azarian, ma3@cedar.evansville.edu
%E A029652 More terms from dww
%I A009001
%S A009001 1,1,1,3,1,5,1,7,1,9,1,11,1,13,1,15,1,17,1,19,1,21,1,23,
%T A009001 1,25,1,27,1,29,1,31,1,33,1,35,1,37,1,39,1,41,1,43,1,45,
%U A009001 1,47,1,49,1,51,1,53,1,55,1,57,1,59,1,61,1,63,1,65,1,67
%V A009001 1,1,-1,-3,1,5,-1,-7,1,9,-1,-11,1,13,-1,-15,1,17,-1,-19,1,21,-1,-23,
%W A009001 1,25,-1,-27,1,29,-1,-31,1,33,-1,-35,1,37,-1,-39,1,41,-1,-43,1,45,
%X A009001 -1,-47,1,49,-1,-51,1,53,-1,-55,1,57,-1,-59,1,61,-1,-63,1,65,-1,-67
%N A009001 Expansion of (1+x)*cos(x); also continued fraction for tan(1).
%H A009001 Index entries for continued fractions for constants
%F A009001 (-1)^(n/2) if n even, n*(-1)^((n-1)/2) if n odd.
%F A009001 a(n) =(n^n mod (n+1))*(-1)^[n/2] for n>0 =(-1)^n*(a(n-2)-a(n-1))-a(n-3) for n>2. - Henry Bottomley (se16@btinternet.com), Oct 19 2001
%t A009001 Cos[ x ]*(1+x)
%Y A009001 Cf. A009531.
%K A009001 sign,done,easy,nice
%O A009001 0,4
%A A009001 rhh@research.bell-labs.com, njas, sp, dww
%E A009001 Formula corrected 03/97 by Olivier Gerard
%I A029669
%S A029669 1,1,3,1,5,1,7,9,5,1,9,1,11,25,7,1,13,55,27,1,15,49,91,105,77,35,9,1,
%T A029669 17,1,19,81,11,1,21,285,65,1,23,121,385,825,275,77,13,1,25,2079,935,1,
%U A029669 27,169,3289,4719,2717,1287,15,1,29,819,5005,9867,7007,1729,119,1,31
%N A029669 Odd numbers in the (2,1)-Pascal triangle A029653.
%K A029669 nonn,tabf
%O A029669 0,3
%A A029669 Mohammad K. Azarian, ma3@cedar.evansville.edu
%E A029669 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A050329
%S A050329 1,1,1,3,1,5,1,7,13,1,13,9,31,1,25,11,57,1,41,101,13,75,63,91,1,61,239,
%T A050329 15,233,129,133,1,85,469,17,535,231,409,183,1,705,113,919,321,815,19,
%U A050329 1029,377,1177,241,1,1671,145,541,2593,681,1301,21,1763,575,2741
%N A050329 Ordered factorizations into square-free factors indexed by prime signatures. A050328(A025487).
%K A050329 nonn
%O A050329 1,4
%A A050329 Christian G. Bower (bowerc@usa.net), Oct 1999.
%I A051707
%S A051707 1,1,1,3,1,5,1,8,3,5,1,21,1,5,5,23,1,21,1,21
%N A051707 Number of factorizations of (n,n) into pairs (k,l).
%C A051707 Pairs (k,l) must satisfy 0t-1 of Z, folded to N.
%C A065168 This permutation consists of just one cycle, which is infinite.
%H A065168 Index entries for sequences that are permutations of the natural numbers
%F A065168 Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then a(n) = f(g(n)-1).
%Y A065168 Inverse permutation to A065164.
%Y A065168 Obtained by composing permutations A065190 and A014681.
%K A065168 nonn
%O A065168 1,1
%A A065168 Antti.Karttunen@iki.fi Oct 19 2001
%I A065277
%S A065277 3,1,5,2,9,7,11,4,17,6,19,13,21,15,23,8,33,10,35,12,37,14,39,25,41,27,
%T A065277 43,29,45,31,47,16,65,18,67,20,69,22,71,24,73,26,75,28,77,30,79,49,81,
%U A065277 51,83,53,85,55,87,57,89,59,91,61,93,63,95,32,129,34,131,36,133,38,135
%N A065277 A065275 conjugated with A059893, inverse of A065278.
%H A065277 Index entries for sequences that are permutations of the natural numbers
%F A065277 a(n) = A059893(A065275(A059893(n)))
%K A065277 nonn
%O A065277 1,1
%A A065277 Antti.Karttunen@iki.fi Oct 28 2001
%I A059971
%S A059971 1,3,1,5,2,13,12,14,13,1,6,13,8,13,1,17,8,158,155,72,170,198,48,145,
%T A059971 208,165,25,55,205,171,206,55,158,6,140,151,53,113,252,191,254,228,26,
%U A059971 116,130,146,243,145,118,72,14,75,115,20,69,60,177,121,99,171,169,170
%N A059971 n^n using Nim multiplication.
%Y A059971 A058734, A006042, A051917.
%K A059971 nonn
%O A059971 1,2
%A A059971 John W. Layman (layman@math.vt.edu), Mar 05 2001
%I A060439
%S A060439 1,3,1,5,3,1,9,3,3,1,27,8,3,3,1
%N A060439 Triangle T(n,k), 1 <= k <= n, giving maximal size of ternary code of length n and covering radius k.
%D A060439 G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 174.
%H A060439 Index entries for sequences related to covering codes
%e A060439 1; 3,1; 5,3,1; 9,3,3,1; ...
%Y A060439 First column gives A004044. Cf. A060438, A060440.
%K A060439 nonn,tabl,nice
%O A060439 1,2
%A A060439 njas, Apr 07 2001
%E A060439 Next term, T(6,1), is in range 63-73.
%I A013597
%S A013597 1,1,1,3,1,5,3,3,1,9,7,5,3,17,27,3,1,29,3,21,7,17,15,9,43,35,15,29,
%T A013597 3,11,3,11,15,17,25,53,31,9,7,23,15,27,15,29,7,59,15,5,21,69,55,21,
%U A013597 21,5,159,3,81,9,69,131,33,15,135,29,13,131,9,3,33,29,25,11,15,29
%N A013597 nextprime(2^n)-2^n.
%H A013597 Table for large n
%p A013597 seq(nextprime(2^i)-2^i,i=0..100);
%Y A013597 Cf. A014210.
%K A013597 nonn
%O A013597 0,4
%A A013597 James Kilfiger (mapdn@csv.warwick.ac.uk)
%I A053641
%S A053641 1,1,3,1,5,3,7,1,9,3,11,5,13,7,15,1,17,3,19,5,21,7,23,9,25,11,27,13,29,
%T A053641 15,31,1,33,3,35,5,37,7,39,9,41,11,43,13,45,15,47,17,49,19,51,21,53,23,
%U A053641 55,25,57,27,59,29,61,31,63,1
%N A053641 Rotate one binary digit to the right, calculate, then rotate one binary digit to the left.
%F A053641 a(n) = A006257(A038572(n)), =n if n odd, =n-2^k+1 if n even and 2^k<=n<2^(k+1)
%e A053641 a(60)=29 because starting with 111100 the right rotation produces 011110 written as 11110 (i.e. 30) and the left rotation produces 11101 (i.e. 29)
%Y A053641 Cf. A006257, A038572.
%K A053641 nonn
%O A053641 1,3
%A A053641 Henry Bottomley (se16@btinternet.com), Mar 22 2000
%I A036233
%S A036233 1,1,3,1,5,3,7,1,9,5,5,3,5,7,3,1,5,9,7,20,3,20,1,3,5,20,9,1,7,3,31,5,3,
%T A036233 20,31,9,5,1,3,7,20,42,31,7,9,20,5,42,1,31,3,7,53,9,31,20,3,5,7,42,53,
%U A036233 1,9,64,31,42,20,53,3,1,5,9,7,64,75,31,1,42,5,20,9,53,7,3,64,86,75,20
%N A036233 Inverse Colombian function.
%C A036233 Contains only Self-Numbers, see A003052.
%F A036233 a(n) = the smallest x with n in the digit summing sequence starting with x
%Y A036233 Cf. A004207, A016052, A007618, A006507, A016052.
%K A036233 easy,nonn
%O A036233 1,3
%A A036233 Miklos SZABO (mike@ludens.hu)
%I A000265 M2222 N0881
%S A000265 1,1,3,1,5,3,7,1,9,5,11,3,13,7,15,1,17,9,19,5,21,11,23,3,25,13,27,7,29,
%T A000265 15,31,1,33,17,35,9,37,19,39,5,41,21,43,11,45,23,47,3,49,25,51,13,53,
%U A000265 27,55,7,57,29,59,15,61,31,63,1,65,33,67,17,69,35,71,9,73,37,75,19,77
%N A000265 Remove 2s from n; or largest odd divisor of n.
%C A000265 When n>0 is written as k*2^j with k odd then k=A000265(n) and j=A007814(n), so: when n is written as k*2^j-1 with k odd then k=A000265(n+1) and j=A007814(n+1), when n>1 is written as k*2^j+1 with k odd then k=A000265(n-1) and j=A007814(n-1)
%D A000265 Problem H-81, Fib. Quart., 6 (1968), 52.
%H A000265 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A000265 a(n) = n/A006519(n) = 2*A025480(n-1)+1
%F A000265 Multiplicative with a(p^e) = 1 if p = 2, p^e if p > 2. - dww, Aug 01, 2001.
%t A000265 Table[ Times@@(#[ [ 1 ] ] ^#[ [ 2 ] ]&/@ Select[ FactorInteger[ i ],#[ [ 1 ] ]\[ NotEqual ]2& ]),{i,90} ] (from Harvey Dale)
%K A000265 nonn,easy,nice
%O A000265 1,3
%A A000265 njas
%E A000265 Additional comments from Henry Bottomley (se16@btinternet.com), Mar 02 2000. More terms from Larry Reeves (larryr@acm.org), Mar 14 2000.
%I A040026
%S A040026 1,1,3,1,5,3,7,1,9,5,11,3,13,7,15,1,17,9,19,5,21,11,23,9,25,13,27,7,29,15,
%T A040026 31,1,33,17,35,9,37,19,39,15,41,21,43,11,45,23,47,15,49,25,51,13,53,
%U A040026 27,55,7,57,29,59,15,61,31,63,1,65,33,67,17,69,35,71,9,73,37,75,19
%V A040026 1,1,3,1,5,3,7,1,9,5,11,-3,13,7,15,1,17,9,19,5,21,11,23,9,25,13,27,-7,29,15,
%W A040026 31,1,33,17,35,9,37,19,39,-15,41,21,43,-11,45,23,47,-15,49,25,51,13,53,
%X A040026 27,55,-7,57,29,59,-15,61,31,63,1,65,33,67,17,69,35,71,9,73,37,75,-19
%N A040026 If n=r*2^s, r odd, then a(n)=t*r, where t is smallest (in magnitude) number such that 1=t*r+u*2^s.
%D A040026 J. Neukirch, Class Field Theory, Springer, p. 1.
%e A040026 24=3*2^3, 1=3*3-1*2^3, a(24)=3*3=9.
%o A040026 (C:) for(n=1;n<=100;n++) { r=n; s=1; while((r&1)==0) { r>>=1; s<<=1; } for(t=1;t<9999;t++) { if(((t*r-1)%s)==0) { printf("%d,",t*r); break; } if(((t*r+1)%s)==0) { printf("%d,",-t*r); break; } } if((n%10)==0) printf("\n"); if(t==9999) exit(0); //"not found": error }
%K A040026 sign,done,easy,nice
%O A040026 1,3
%A A040026 njas
%E A040026 More terms from Arlin Anderson (arlin@myself.com)
%I A030101
%S A030101 0,1,1,3,1,5,3,7,1,9,5,13,3,11,7,15,1,17,9,25,5,21,13,29,3,19,11,27,7,23,
%T A030101 15,31,1,33,17,49,9,41,25,57,5,37,21,53,13,45,29,61,3,35,19,51,11,43,27,
%U A030101 59,7,39,23,55,15,47,31,63,1,65,33,97,17,81,49,113,9,73,41,105,25,89,57
%N A030101 Base 2 reversal of n (written in base 10).
%D A030101 Solutions to 17th USA Mat. Olympiad, Math. Mag., 62 (1989), 210-214 (#3).
%Y A030101 Cf. A030109.
%K A030101 nonn,base,nice
%O A030101 0,4
%A A030101 dww
%I A060819
%S A060819 1,1,3,1,5,3,7,2,9,5,11,3,13,7,15,4,17,9,19,5,21,11,23,6,25,13,27,7,29,
%T A060819 15,31,8,33,17,35,9,37,19,39,10,41,21,43,11,45,23,47,12,49,25,51,13,53,
%U A060819 27,55,14,57,29,59,15,61,31,63,16,65,33,67,17,69,35,71,18,73,37,75,19
%N A060819 a(n) = n / gcd(n,4).
%F A060819 G.f.: (x^7+x^6+3x^5+x^4+3x^3+x^2+x)/(1-x^4)^2
%Y A060819 Cf. A026741, A051176, A060791, A060789.
%K A060819 nonn,easy
%O A060819 1,3
%A A060819 Len Smiley (smiley@math.uaa.alaska.edu), Apr 30 2001
%E A060819 More terms from Larry Reeves (larryr@acm.org), May 07 2001
%I A002323 M2223 N0882
%S A002323 1,3,1,5,3,15,3,20,1,1,1,32,37,22,36,8,36,10,1,7,49,48,23,77,92,81,
%T A002323 13,95,49,1,17,95,30,96,66,132,67,107,3,50,148,25,52,175,167,109,143,201
%N A002323 Fermat remainders.
%D A002323 W. Meissner, Ueber die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte K\"{o}niglich Preussischen Akadamie Wissenschaften Berlin, 35 (1913), 663-667.
%D A002323 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
%Y A002323 Cf. A001917.
%K A002323 nonn
%O A002323 3,2
%A A002323 njas
%I A029655
%S A029655 1,1,3,1,5,4,1,7,9,5,1,9,16,14,6,1,11,25,30,20,7,1,13,36,55,50,27,8,1,
%T A029655 15,49,91,105,77,35,9,1,17,64,140,196,182,112,44,10,1,19,81,204,336,
%U A029655 378,294,156,54,11,1,21,100,285,540,714,672,450,210,65,12,1,23,121,385
%N A029655 Numbers in the (2,1)-Pascal triangle A029653 that are different from 2.
%K A029655 nonn,tabf
%O A029655 0,3
%A A029655 Mohammad K. Azarian, ma3@cedar.evansville.edu
%E A029655 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A016574
%S A016574 3,1,5,4,2,1,2,7,4,1,1,12,22,1,2,2,6,16,10,1,2,129,1,1,
%T A016574 7,1,96,1,2,3,8,1,2,1,5,5,4,1,7,2,1,1,406,1,18,1,1,2,1,
%U A016574 17,1,2,3,15,1,1,1,1,3,6,1,1,20,1,1,1,1,5,1,11,1,1,1,1
%N A016574 Continued fraction for ln(93/2).
%K A016574 nonn,cofr
%O A016574 1,1
%A A016574 njas
%I A011090
%S A011090 3,1,5,4,3,4,2,1,4,5,5,2,9,9,0,4,2,3,4,9,2,3,1,2,6,8,4,9,2,9,4,6,8,
%T A011090 7,9,8,0,4,9,6,2,3,6,2,9,3,3,0,9,8,3,5,7,8,9,9,5,4,1,0,1,7,5,8,1,9,
%U A011090 9,1,8,5,5,2,6,8,7,2,4,9,1,2,1,5,0,8,2,8,2,5,4,3,8,9,2,2,8,0,2,3,4
%N A011090 Decimal expansion of 4th root of 99.
%K A011090 nonn,cons
%O A011090 1,1
%A A011090 njas
%I A021321
%S A021321 0,0,3,1,5,4,5,7,4,1,3,2,4,9,2,1,1,3,5,6,4,6,6,8,7,6,9,7,1,6,0,8,8,
%T A021321 3,2,8,0,7,5,7,0,9,7,7,9,1,7,9,8,1,0,7,2,5,5,5,2,0,5,0,4,7,3,1,8,6,
%U A021321 1,1,9,8,7,3,8,1,7,0,3,4,7,0,0,3,1,5,4,5,7,4,1,3,2,4,9,2,1,1,3,5,6
%N A021321 Decimal expansion of 1/317.
%K A021321 nonn,cons
%O A021321 0,3
%A A021321 njas
%I A010847
%S A010847 0,0,1,1,3,1,5,4,6,4,9,4,11,8,10,11,15,8,17,12,16,15,21,13,22,19,23,
%T A010847 20,27,12,29,26,27,26,30,22,35,30,33,29,39,23,41,35,37,38,45,33,46,38,
%U A010847 45,43,51,38,50,45,51,50,57,34,59,54,55,57,60,44,65,58,63,50,69,54,71
%N A010847 Number of numbers <= n with a prime factor not in n.
%K A010847 nonn,easy
%O A010847 1,5
%A A010847 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A007085 M2224
%S A007085 0,0,0,0,0,0,0,1,0,0,1,3,1,5,4,11,20,46
%N A007085 Trivalent 3-connected bipartite planar graphs with 4n nodes.
%D A007085 D. A. Holton et al., Hamiltonian cycles in cubic 3-connected bipartite planar graphs, J. Combin. Theory, B 38 (1985), 279-297.
%K A007085 nonn
%O A007085 2,12
%A A007085 njas, mb
%I A049266
%S A049266 1,1,3,1,5,6,0,1,2,3,1,5,5,7,8,1,5,2,3,5,14,1,1,17,2,19,20,0,1,2,3,4,5,
%T A049266 3,7,8,1,10,8,12,13,14,1,1,5,9,5,17,21,2,11,5,1,17,3,28,29,1,13,32,33,
%U A049266 34,0,1,2,3,1,5,6,7,1,8,5,1,3,12,14,2,1,17,2,19,5,21,22,1,5,1,13,17,7
%N A049266 Smallest nonnegative value taken on by nx^2 - 7y^2 for an infinite number of integer pairs (x, y).
%K A049266 nonn
%O A049266 1,3
%A A049266 dww
%I A016600
%S A016600 3,1,5,7,0,0,0,4,2,1,1,5,0,1,1,3,2,7,7,4,0,3,7,1,8,5,4,8,3,1,3,9,9,
%T A016600 7,1,4,0,8,2,0,5,5,0,3,6,7,6,5,9,9,6,8,7,7,9,0,7,9,8,2,8,3,3,7,3,1,
%U A016600 3,4,2,4,5,9,1,5,3,6,1,0,6,3,5,7,0,0,4,8,2,6,4,2,6,4,3,8,3,2,7,2,9
%N A016600 Decimal expansion of ln(47/2).
%K A016600 nonn,cons
%O A016600 1,1
%A A016600 njas
%I A038871
%S A038871 0,1,1,3,1,5,7,1,5,11,15,1,9,21,27,31,1,9,21,43,55,63,1,17,37,85,91,
%T A038871 119,127,1,17,73,85,171,219,239,255,1,33,73,165,341,363,439,495,511,
%U A038871 1,33,137,293,341,683,731,887,991,1023,1,65,273,585,661,1365,1387
%N A038871 Orbits of order k under doubling map which remain in a semicircle, with k dividing n.
%Y A038871 Cf. A038870.
%K A038871 nonn,tabl,easy
%O A038871 1,4
%A A038871 Francois Maurel (maurel@sequoia.ens.fr)
%I A001607 M2225 N0883
%S A001607 0,1,1,1,3,1,5,7,3,17,11,23,45,1,91,89,93,271,85,457,627,287,
%T A001607 1541,967,2115,4049,181,8279,7917,8641,24475,7193,41757,
%U A001607 56143,27371,139657,84915,194399,364229,24569,753027,703889
%V A001607 0,1,-1,-1,3,-1,-5,7,3,-17,11,23,-45,-1,91,-89,-93,271,-85,-457,627,287,
%W A001607 -1541,967,2115,-4049,-181,8279,-7917,-8641,24475,-7193,-41757,
%X A001607 56143,27371,-139657,84915,194399,-364229,-24569,753027,-703889
%N A001607 a(n) = -a(n-1) - 2a(n-2).
%D A001607 Problem E2367, Amer. Math. Monthly, 79 (1972), 772.
%D A001607 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
%K A001607 sign,done,easy
%O A001607 0,5
%A A001607 njas
%I A021080
%S A021080 0,1,3,1,5,7,8,9,4,7,3,6,8,4,2,1,0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,
%T A021080 1,0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,1,0,5,2,6,3,1,5,7,8,9,4,7,3,6,
%U A021080 8,4,2,1,0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,1,0,5,2,6,3,1,5,7,8,9,4
%N A021080 Decimal expansion of 1/76.
%K A021080 nonn,cons
%O A021080 0,3
%A A021080 njas
%I A049764
%S A049764 0,0,1,1,3,1,5,7,9,10,18,10,18,25,28,40,44,26,52,79,60,78,73,101,117,
%T A049764 133,114,136,91,90,158,188,220,244,218,220,208,283,280,303,220,319,287,
%U A049764 393,398,446,391,459,435,534,481,471,428,499
%N A049764 a(n)=SUM{T(n,k): k=1,2,...,n}, array T as in A049763.
%K A049764 nonn
%O A049764 1,5
%A A049764 Clark Kimberling, ck6@cedar.evansville.edu
%I A038738
%S A038738 1,1,3,1,5,8,1,7,17,21,1,9,30,50,55,1,11,47,103,138,144,1,13,68,188,
%T A038738 314,370,377,1,15,93,313,643,895,979,987,1,17,122,486,1201,1993,2455,
%U A038738 2575,2584,1,19,155,715,2080,4082,5798,6590,6755
%N A038738 Path-counting array T(i,j) obtained from array t in A038792 by T(i,j)=t(2i+1,j).
%C A038738 T(n,n)=A001906(n) for n >= 0 (even-indexed Fibonacci numbers).
%C A038738 Row sums: A030267.
%e A038738 Rows: {1}; {1,3}; {1,5,8}; {1,7,17,21}; ...
%K A038738 nonn,tabl
%O A038738 1,3
%A A038738 Clark Kimberling, ck6@cedar.evansville.edu, May 02 2000
%I A063858
%S A063858 1,1,3,1,5,8,1,7,27,46,1,9,71,1063,790,1,11,156,44956,1434219,37829,1,13,325
%N A063858 Triangle T(n,k) (n >= 2, k = 0..n-2) giving number of abstract dissection types of configurations of n points in n-k dimensions.
%H A063858 L. Finschi, Homepage of Oriented Matroids
%H A063858 L. Finschi and K. Fukuda, Complete combinatorial generation of small point set configurations and hyperplane arrangements, pp. 97-100 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%e A063858 1; 1,3; 1,5,8; 1,7,27,46; ...
%Y A063858 Diagonals give A063859, A063860, A063861. Row sums give A063862.
%K A063858 nonn,tabl,nice
%O A063858 2,3
%A A063858 njas, Aug 26 2001
%I A029723
%S A029723 1,1,3,1,5,9,3,1,21,7,13,13,5
%N A029723 Trace of Frobenius of the reduction mod 2 of the elliptic curve C / L, L a lattice with Gram matrix [ 4 1; 1 2n ].
%F A029723 Of form sqrt(2^(h(1-8n)+2)-m^2(8n-1)), with m odd.
%K A029723 nonn
%O A029723 0,3
%A A029723 Eric Rains (rains@research.att.com)
%I A055199
%S A055199 3,1,5,9,9,8,6,1,7,2
%N A055199 10^n-th decimal digit of Pi.
%D A055199 David Blatner, The Joy of Pi, Walker and Co., NY, 1997; page 130 mentions the millionth digit.
%H A055199 GJ Technologies, Up to 10000000 digits of Pi
%H A055199 Yasumasa Kanada, 6,442,450,000 decimal digits of pi and 1/pi
%H A055199 Aoki Mitsuru, Multiple digits of Pi
%H A055199 Steve Pagliarulo, Quick Pi
%H A055199 Daniel Sedory, The Pi Pages
%H A055199 Daniel Sedory, 100,000 Decimal Digits of Pi
%e A055199 3 is the -1st decimal digit of Pi, 1 is the 1st decimal digit of Pi, 5 is the 10th decimal digit of Pi, etc.
%t A055199 Do[Print[IntegerPart[Mod[N[Pi*10^(10^n),10^n+25],10]]],{n,-1,7}]
%Y A055199 Cf. A000796.
%K A055199 nonn,base
%O A055199 -1,1
%A A055199 Robert G. Wilson v (RGWv@kspaint.com), Jun 30 2000
%I A065229
%S A065229 1,1,3,1,5,10,4,11,19,6,16,27,4,17,31,46,11,28,46,65,15,36,58,81,13,38,
%T A065229 64,91,2,31,61,92,124,12,46,81,117,154,16,55,95,136,178,11,55,100,146,
%U A065229 193,241,43,93,144,196,249,16,71,127,184,242,301,31,92,154,217,281,346
%N A065229 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the pentagonal numbers. The first elements of the rows form a(n).
%Y A065229 Cf. A064766, A064865, A065221-A065228, A065230-A065223.
%K A065229 easy,nonn
%O A065229 0,3
%A A065229 Floor van Lamoen (f.v.lamoen@wxs.nl), Oct 22 2001
%I A063853
%S A063853 1,1,3,1,5,11,1,8,55,93,1,11,204,5083,2121,1,15,705,505336,10775236,
%T A063853 122508,1,19,2293
%N A063853 Triangle T(n,k) (n >= 3, k = 1..n-2) giving number of abstract order types of configurations of n points in n-k dimensions.
%H A063853 L. Finschi, Homepage of Oriented Matroids
%H A063853 L. Finschi and K. Fukuda, Complete combinatorial generation of small point set configurations and hyperplane arrangements, pp. 97-100 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%e A063853 1; 1,3; 1,5,11; 1,8,55,93; ...
%Y A063853 Diagonals give A063854, A063855, A063856. Row sums give A063857.
%K A063853 nonn,tabl,nice
%O A063853 3,3
%A A063853 njas, Aug 26 2001
%I A059616
%S A059616 0,0,1,3,1,5,15,7,7,9,15,55,33,13,91,105,10,34,153,57,95,105,77,
%T A059616 253,69,25,325,351,63,203,435,155,124,132,187,595,315,111,703,741,
%U A059616 65,205,861,301,473,495,345,1081,282,98,1225,1275,221,689,1431
%N A059616 Numerator of (n*(n-1)/(8*(2*n+1)).
%Y A059616 Cf. A059617.
%K A059616 nonn,frac
%O A059616 0,4
%A A059616 njas, Feb 18 2001
%I A039512
%S A039512 1,3,1,5,19,23,187,347,5,7,9,1,11,13,1,13,131,211,227,251,259,283,287,
%T A039512 319,3,15,57,69,561,1041,1,17,23,5,19,15,21,5,7,23,41,5,7,17,25,95,115,
%U A039512 935,1735,27,1,11,29,3811,7055,13,31,3,33,39,7,13,17,25,35,133
%N A039512 T(n,1..i) are attractors in '3x+(2n+1)' problem.
%e A039512 1; 3; 1,5,19,23,187,347; 5,7; 9; 1,11,13;...
%Y A039512 A039508 gives number of columns in each row. Cf. A039508-A039515.
%K A039512 nonn,tabf
%O A039512 0,2
%A A039512 Christian G. Bower (bowerc@usa.net), Feb 1999.
%I A037280
%S A037280 1,1,3,1,5,23,7,3,3,25,11,1173,13,27,35,31,17,2369,19,12255,37,211,23,
%T A037280 586703,5,213,39,12357,29,23561015,31,1551,311,217,57,117345609,37,219,
%U A037280 313,6145255,41,23671421,43,120561,35915,223,47,2933515203,7,251025
%N A037280 a(2)=1; if n>2 composite replace by concatenation of its nontrivial divisors [ A037279 ] then divide out any factors of 2.
%e A037280 Divisors of 12 are 1,2,3,4,6,12, so 12->2346=2*1173->1173.
%Y A037280 Cf. A037279.
%K A037280 nonn,easy,base
%O A037280 0,3
%A A037280 njas
%E A037280 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A011002
%S A011002 1,3,1,6,0,7,4,0,1,2,9,5,2,4,9,2,4,6,0,8,1,9,2,1,8,9,0,1,7,9,6,9,9,
%T A011002 9,0,5,5,1,6,0,0,6,8,5,9,0,2,0,5,8,2,2,1,7,6,7,3,1,9,2,2,6,5,8,5,9,
%U A011002 5,8,6,6,7,9,5,1,9,7,3,0,2,1,3,3,0,5,0,7,4,3,1,5,0,2,4,6,6,0,1,9,3
%N A011002 Decimal expansion of 4th root of 3.
%K A011002 nonn,cons
%O A011002 1,2
%A A011002 njas
%I A034839
%S A034839 1,1,1,1,1,3,1,6,1,1,10,5,1,15,15,1,1,21,35,7,1,28,70,28,1,1,36,126,
%T A034839 84,9,1,45,210,210,45,1,1,55,330,462,165,11,1,66,495,924,495,66,1,1,
%U A034839 78,715,1716,1287,286,13
%N A034839 Triangular array formed by taking every other term of each row of Pascal's triangle.
%e A034839 1; 1; 1, 1; 1, 3; 1, 6, 1; 1, 10, 5; 1, 15, 15, 1;...
%Y A034839 Cf. A007318.
%K A034839 nonn,easy,tabf
%O A034839 0,6
%A A034839 njas
%I A010287
%S A010287 3,1,6,1,2,1,10,1,2,7,1,4,15,37,1,6,1,2,2,1,2,8,10,39,75,
%T A010287 1,2,2,1,1,2,1,10,18,1,4,3,2,1,2,1,2,1,5,1,4,1,1,3,181,
%U A010287 1,2,1,5,11,2,16,1,2,1,2,3,1,1,1,3,3,1,1,15,3,8,3,3,2,2
%N A010287 Continued fraction for cube root of 58.
%K A010287 nonn,cofr
%O A010287 0,1
%A A010287 njas
%I A016476
%S A016476 3,1,6,1,3,4,4,1,13,2,4,1,22,3,2,7,3,4,1,3,4,7,11,2,1,39,
%T A016476 1,1,1,7,8,3,1,5,4,2,6,1,1,4,35,2,45,2,1,1,6,2,1,1,3,1,
%U A016476 3,1,13,1,3,3,26,2,17,1,1,3,1,45,1,5,1,1,1,3,2,2,1,3,4
%N A016476 Continued fraction for ln(48).
%K A016476 nonn,cofr
%O A016476 1,1
%A A016476 njas
%I A019570
%S A019570 0,3,1,6,1,5,2
%V A019570 0,3,1,6,-1,-5,2
%N A019570 Zeroth row of infinite Latin square.
%D A019570 M. J. Caulfield, Full and quarter plane complete infinite Latin squares, Discr. Math., 159 (1996), 251-253.
%H A019570 Index entries for sequences related to Latin squares and rectangles
%Y A019570 Cf. A019585.
%K A019570 nonn,sign,done,more
%O A019570 0,2
%A A019570 njas
%I A040011
%S A040011 3,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,
%T A040011 1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,
%U A040011 6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,6
%N A040011 Continued fraction for sqrt(15).
%H A040011 Index entries for continued fractions for constants
%p A040011 Digits:=100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
%K A040011 nonn,cofr,easy
%O A040011 0,1
%A A040011 njas
%I A001065 M2226 N0884
%S A001065 0,1,1,3,1,6,1,7,4,8,1,16,1,10,9,15,1,21,1,22,11,14,1,36,6,16,13,28,1,
%T A001065 42,1,31,15,20,13,55,1,22,17,50,1,54,1,40,33,26,1,76,8,43,21,46,1,66,17,
%U A001065 64,23,32,1,108,1,34,41,63,19,78,1,58,27,74,1,123,1,40,49,64,19,90,1,106
%N A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.
%D A001065 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A001065 George E. Andrews, Number Theory. New York: Dover, 1994 . Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
%H A001065 H. Bottomley, Illustration of initial terms)
%H A001065 E. W. Weisstein, Link to a section of The World of Mathematics (1).
%H A001065 E. W. Weisstein, Link to a section of The World of Mathematics (2).
%H A001065 Index entries for "core" sequences
%e A001065 For n=44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
%p A001065 with(numtheory); [ seq(sigma(n)-n,n=1..100) ];
%t A001065 Table[ Plus @@ Select[ Divisors[ n ], # 3, 11ab..yz -> 11ab..yz0, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11ab..y1.
%C A065275 On the right side every node replaces its left child, on the left side the left children replace their parents, and the right children are transferred to the same offset at the right side (staying right children). See comment at A065263.
%H A065275 Index entries for sequences that are permutations of the natural numbers
%p A065275 RightChildTransferred := proc(n) local k; if(1 = n) then RETURN(3); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN(2*n); fi; if(0 = (n mod 2)) then RETURN(n/2); fi; RETURN(n + (2^k)); end;
%Y A065275 A057114, A065263, A065269, A065281, A065287. Inverse: A065276, conjugated with A059893: A065277, and the inverse of that: A065278.
%K A065275 nonn
%O A065275 1,1
%A A065275 Antti.Karttunen@iki.fi Oct 28 2001
%I A065567
%S A065567 1,3,1,6,3,1,10,7,4,1,15,11,10,5,1,21,20,21,15,6,1,28,26,36,35,21,7,1,
%T A065567 36,38,60,71,56,28,8,1,45,50,90,127,126,84,36,9,1,55,67,132,215,253,
%U A065567 210,120,45,10,1,66,77,177,335,463,462,330,165,55,11,1,78,105,250,512
%N A065567 T(n,m) is the sum over all m-subsets of {1,..,n} of the GCD of the subset.
%C A065567 First differences of row sums equals A034738.
%e A065567 1; 3,1; 6,3,1; 10,7,4,1; 15,11,10,5,1; ...
%e A065567 T(4,2)=7 since GCD[1,2] + GCD[1,3] + GCD[1,4] + GCD[2,3] + GCD[2,4] + GCD[3,4] = 7.
%t A065567 Table[Plus@@(GCD@@@KSubsets[Range[n],m]),{n,16},{m,n}]
%Y A065567 Cf. A065568, A034738.
%K A065567 nonn,tabl
%O A065567 1,2
%A A065567 Wouter Meeussen (wouter.meeussen@vandemoortele.com), Nov 30 2001
%I A039805
%S A039805 1,3,1,6,3,1,13,9,3,1,23,19,9,3,1,44,42,22,9,3,1,74,80,48,22,9,3,1,
%T A039805 129,154,99,51,22,9,3,1,210,273,193,105,51,22,9,3,1,345,484,362,212,
%U A039805 108,51,22,9,3,1,542,815,651,401,218,108,51,22,9,3,1,858,1369,1147
%N A039805 Matrix cube of partition triangle A008284.
%e A039805 1; 3,1; 6,3,1; 13,9,3,1; ...
%Y A039805 Cf. A038497, A038498, A039806, A039807. a(n, 1) = A022811(n) (first column).
%K A039805 nonn,tabl
%O A039805 1,2
%A A039805 Christian G. Bower (bowerc@usa.net), Feb 1999.
%I A008795
%S A008795 1,0,3,1,6,3,10,6,15,10,21,15,28,21,36,28,45,36,55,45,66,
%T A008795 55,78,66,91,78,105,91,120,105,136,120,153,136,171,153,
%U A008795 190,171,210,190,231,210,253,231,276,253,300,276,325,300
%N A008795 Molien series for 3-dimensional group [ 2,n ]+ = 22n.
%p A008795 (1+x^3)/(1-x^2)^3
%K A008795 nonn,easy
%O A008795 0,3
%A A008795 njas
%I A021320
%S A021320 0,0,3,1,6,4,5,5,6,9,6,2,0,2,5,3,1,6,4,5,5,6,9,6,2,0,2,5,3,1,6,4,5,
%T A021320 5,6,9,6,2,0,2,5,3,1,6,4,5,5,6,9,6,2,0,2,5,3,1,6,4,5,5,6,9,6,2,0,2,
%U A021320 5,3,1,6,4,5,5,6,9,6,2,0,2,5,3,1,6,4,5,5,6,9,6,2,0,2,5,3,1,6,4,5,5
%N A021320 Decimal expansion of 1/316.
%K A021320 nonn,cons
%O A021320 0,3
%A A021320 njas
%I A007383 M2228
%S A007383 0,0,1,0,3,1,6,4,11,10,20,21,36,41,64,77,113,141,199,254,350,453,615,803,
%T A007383 1080,1418,1896,2498,3328,4394,5841,7722,10251,13563,17990,23814,31571
%N A007383 Strict 1st-order maximal independent sets in path graph.
%D A007383 R. Yanco and A. Bagchi, ``K-th order maximal independent sets in path and cycle graphs,'' J. Graph Theory, submitted, 1994.
%K A007383 nonn
%O A007383 1,5
%A A007383 njas,mb
%I A016575
%S A016575 3,1,6,5,1,1,4,1,5,11,1,1,3,1,2,1,2,1,1,1,19,2,1,1,1,3,
%T A016575 2,8,5,1,1,1,1,4,6,1,1,2,191,4,1,14,1,1,1,1,17,2,4,9,1,
%U A016575 2,2,3,1,31,2,2,1,3,3,1,21,101,2,1,1,1,3,2,2,7,6,2,6,14
%N A016575 Continued fraction for ln(95/2).
%K A016575 nonn,cofr
%O A016575 1,1
%A A016575 njas
%I A061702
%S A061702 0,1,3,1,6,5,1,9,18,6,1,12,42,44,9,1,15,75,145,95,13,1,18,117,336,420,
%T A061702 192,20,1,21,168,644,1225,1085,371,31,1,24,228,1096,2834,3880,2588,696,
%U A061702 49,1,27,297,1719,5652,10656,11097,5823,1278,78,1,30,375,2540,10165
%N A061702 Triangle T(n,k) defined by Sum_{n >= 0,m >= 0} T(n,m)*x^n*y^m = y*(1 + 3*x - 4*x^2*y - 3*x^2*y^2 - 3*x^3*y^2 + 4*x^4*y^3)/((1 - y - 2*x*y - x*y^2 + x^3*y^3)*(1 - x*y)).
%D A061702 R. P. Stanley, Enumerative Combinatorics I, Example 4.7.17.
%Y A061702 Cf. A000183, row sums: A061703, third column: A000338, fourth column: A000561, fifth column: A000562, sixth column: A000563, seventh column: A000564, eighth column: A000565.
%K A061702 easy,nonn,tabl
%O A061702 0,3
%A A061702 Vladeta Jovovic (vladeta@Eunet.yu), Jun 18 2001
%I A049410
%S A049410 1,3,1,6,9,1,6,51,18,1,0,210,195,30,1,0,630,1575,525,45,1,0,1260,10080,
%T A049410 6825,1155,63,1,0,1260,51660,71505,21840,2226,84,1,0,0,207900,623700,
%U A049410 333585,57456,3906,108,1,0,0,623700,4573800,4293135,1195425,131670
%N A049410 A triangle of numbers related to triangle A049325.
%C A049410 a(n,1)= A008279(3,n-1). a(n,m)=: S1(-3;n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1;n,m)= A008275 (signed Stirling 1st kind), S1(2;n,m)= A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A000369(n,m).
%C A049410 The monic row polynomials E(n,x):=sum(a(n,m)*x^m,m=1..n), E(0,x):=1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%H A049410 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A049410 a(n,m) = n!*A049325(n,m)/(m!*4^(n-m)); a(n,m) = (4*m-n+1)*a(n-1,m) + a(n-1,m-1), n >= m >= 1; a(n,m)=0, n= 3.
%F A025230 G.f.: (1-sqrt(1-12*x+32*x^2))/2 - Michael Somos, June 8, 2000.
%o A025230 (PARI) a(n)=polcoeff((1-sqrt(1-12*x+32*x^2+x*O(x^n)))/2,n)
%K A025230 nonn
%O A025230 1,1
%A A025230 Clark Kimberling (ck6@cedar.evansville.edu)
%I A051511
%S A051511 1,3,1,7,0,0,2,2,25,1,1,1,1,4,0,2,1,24,0,0,0,0,4,1,2,4,1,2,5,1,1,5,0,1,1,1,2,1,1,2,4,1,4,1,10,0,0,6,0,1,160,0,0,0,1,17,0,1,4,4,3,0,0,14,1,7,0,1,2,1,6,0,0,0,16,1,1,1,3390,1,1,1,2,0,5,1,2,0,0,0,0,4,2,1,2,9,6,2,30,0,1,2,0,1,3,1,0,0,1,1,2,1,0,1,2,0,0,2,1,1,1
%V A051511 1,-3,1,-7,0,0,2,-2,25,-1,-1,-1,-1,-4,0,2,-1,-24,0,0,0,0,4,-1,-2,4,-1,-2,5,-1,-1,-5,0,1,1,-1,-2,-1,-1,2,4,-1,4,-1,10,0,0,-6,0,1,-160,0,0,0,1,-17,0,-1,-4,-4,-3,0,0,-14,1,7,0,-1,-2,1,6,0,0,0,16,-1,-1,1,3390,-1,1,-1,-2,0,-5,-1,-2,0,0,0,0,-4,-2,1,-2,9,-6,2,30,0,-1,-2,0,-1,-3,-1,0,0,-1,-1,-2,-1,0,1,2,0,0,-2,-1,1,-1
%N A051511 [tan(2^n)].
%K A051511 sign,done,easy
%O A051511 0,2
%A A051511 njas
%E A051511 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A026499
%S A026499 3,1,7,1,1,3,3,1,8,1,6,1,7,1,1,3,3,1,1,7,1,3,3,1,7,1,1,3,3,1,
%T A026499 8,1,6,1,8,1,3,3,1,1,7,1,6,1,7,1,1,3,3,1,8,1,6,1,7,1,1,1,1,1,
%U A026499 1,1,1,1,1,1,1,1,1,2,3,2,1,2,1,2,3,2,3,2,3,2,1,2,1,2,3,2,1,2
%N A026499 a(n) = length of n-th run of identical symbols in A026498.
%K A026499 nonn
%O A026499 1,1
%A A026499 Clark Kimberling, ck6@cedar.evansville.edu
%I A021991
%S A021991 0,0,1,0,1,3,1,7,1,2,2,5,9,3,7,1,8,3,3,8,3,9,9,1,8,9,4,6,3,0,1,9,2,
%T A021991 5,0,2,5,3,2,9,2,8,0,6,4,8,4,2,9,5,8,4,5,9,9,7,9,7,3,6,5,7,5,4,8,1,
%U A021991 2,5,6,3,3,2,3,2,0,1,6,2,1,0,7,3,9,6,1,4,9,9,4,9,3,4,1,4,3,8,7,0,3
%N A021991 Decimal expansion of 1/987.
%K A021991 nonn,cons
%O A021991 0,6
%A A021991 njas
%I A053381
%S A053381 1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,9,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,11,1,3,
%T A053381 1,7,1,3,1,8,1,3,1,7,1,3,1,9,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,15,1,3,1,7,
%U A053381 1,3,1,8,1,3,1,7,1,3,1,9,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,11,1,3,1,7,1,3
%N A053381 Maximal number of linearly independent smooth nowhere zero vector fields on a (2n+1)-sphere.
%C A053381 The number is 0 if n is even (``you can't comb the hair on a basketball''). The "3" and "7" are due to the quaternions and octonions.
%D A053381 A. Hurwitz: "Uber die Komposition der quadratischen formen, Math. Annalen 88 (1923) 1-25.
%D A053381 J. Radon: Lineare Scharen Orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922) 1-14.
%D A053381 J. Frank Adams: Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39-41; Annals of Math. 75 (1962) 603-632.
%D A053381 M. Kervaire: Non-parallelizability of the sphere for $n > 7$, Proc. Nat'l Acad. Sci. USA 44 (1958) 280-283.
%D A053381 J. Milnor: Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.
%F A053381 Let f(n) be the number of linearly independent smooth nowhere zero vector fields on an n-sphere. Then f(n) = 2^c + 8d - 1 where n+1 = (2a+1) 2^b and b = c+4d and 0 <= c <= 3. f(n) = 0 if n is even.
%p A053381 with(numtheory): for n from 1 to 601 by 2 do c:=irem(ifactors(n+1)[2,1,2],4): d:=iquo(ifactors(n+1)[2,1,2],4): printf(`%d,`, 2^c+8*d-1) od:
%o A053381 (C:) int MaxLinInd(int n){ /* Returns max # linearly indep smooth nowhere zero * vector fields on S^{n-1}, n=1,2,... */ int b,c,d,rho; b = 0; while((n & 1)==0){ n /= 2; b++; } c = b & 3; d = (b - c)/4; rho = (1 << c) + 8*d; return( rho - 1); }
%Y A053381 Cf. A047680, A001676.
%K A053381 nonn,nice,easy
%O A053381 0,2
%A A053381 wds@research.nj.nec.com (Warren Smith), Jan 06 2000
%E A053381 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 01 2000
%I A038712
%S A038712 1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,63,1,
%T A038712 3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,127,1,
%U A038712 3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,63,1,3
%N A038712 n XOR (n-1), i.e. nim-sum of sequential pairs.
%H A038712 Index entries for sequences related to Nim-sums
%F A038712 a(n) = 2^A001511(n)-1 = 2*A006519(n)-1 = 2^(A007814(n)+1)-1
%F A038712 Multiplicative with a(2^e)=2^(e+1)-1, a(e^p)=1, p>2. - Vladeta Jovovic (vladeta@Eunet.yu), Nov 06 2001
%e A038712 a(6) = 3 because 110 XOR 101 = 11 base 2 = 3
%Y A038712 Cf. A038713 translated from binary, diagonals of A003987 either side of main diagonal.
%Y A038712 Cf. also A062383.
%K A038712 easy,nonn,mult
%O A038712 1,2
%A A038712 Henry Bottomley (se16@btinternet.com), May 02 2000
%I A038870
%S A038870 0,1,1,3,1,7,1,5,11,15,1,31,1,9,21,43,55,63,1,37,91,127,1,17,85,171,
%T A038870 239,255,1,73,439,511,1,33,137,293,341,683,731,887,991,1023,1,661,
%U A038870 1387,2047,1,65,273,585,1189,1365,2731,2907,3511,3823,4031,4095,1
%N A038870 Orbits of order exactly n under doubling map which remain in a semicircle.
%C A038870 If alpha=exp(2i*pi*a(d,n)/(2^n - 1)), the orbit of alpha has period n and stays in the semi-circle of minimal argument alpha.
%F A038870 a(d,n)=Sum 2^[ nk/d ], k=0..{d-1}; (d,n)=1.
%Y A038870 Cf. A038871.
%K A038870 nonn,tabl,easy
%O A038870 1,4
%A A038870 Francois Maurel (maurel@sequoia.ens.fr)
%I A063754
%S A063754 0,1,1,3,1,7,1,8,5,11,1,20,1,15,13,20,1,31,1,32,17,23,1,52,9,27,21,44,
%T A063754 1,71,1,48,25,35,21,88,1,39,29,84,1,99,1,68,61,47,1,128,13,83,37,80,1,
%U A063754 123,29,116,41,59,1,200,1,63,81,112,33,155,1,104,49,159,1,228,1,75,101
%N A063754 Dirichlet convolution of totient and cototient.
%F A063754 a(n)= Sum{A000010(d)*A051953(n/d)}, d runs over divisors of n.
%e A063754 n = 24: divisors = {1, 2, 3, 4, 6, 8, 12, 24}, d-Phi[d] = {0, 1, 1, 2, 4, 4, 8, 16}, Phi[n/d] = {8, 4, 4, 2, 2, 2, 1, 1}, products = {0, 4, 4, 4, 8, 8, 8, 16}, a(24) = 52.
%Y A063754 A000005, A051953, A029935.
%K A063754 easy,nonn
%O A063754 0,4
%A A063754 Labos E. (labos@ana1.sote.hu), Aug 14 2001
%I A050521
%S A050521 1,1,1,3,1,7,1,10,4,7,1,33,1,7,9,28,1,39,1,33
%N A050521 Number of factorizations of (n,2*n) into pairs (k,l).
%C A050521 Pairs (k,l) must satisfy 0 3, 11ab..yz -> 11ab..yz1, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11ab..y0.
%C A065287 On the right side every node replaces its right child, on the left side the left children replace their parents, and the right children are transferred to the same offset - 1 at the right side (becoming left children). See comment at A065263.
%H A065287 Index entries for sequences that are permutations of the natural numbers
%p A065287 RightChildTransferred_1 := proc(n) local k; if(1 = n) then RETURN(3); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN((2*n)+1); fi; if(0 = (n mod 2)) then RETURN(n/2); fi; RETURN(n + (2^k) - 1); end;
%Y A065287 A057114, A065263, A065269, A065275, A065281. Inverse: A065288, conjugated with A059893: A065289, and the inverse of that: A065290.
%K A065287 nonn
%O A065287 1,1
%A A065287 Antti.Karttunen@iki.fi Oct 28 2001
%I A065263
%S A065263 3,1,7,2,6,13,15,4,14,5,12,25,27,29,31,8,30,9,28,10,26,11,24,49,51,53,
%T A065263 55,57,59,61,63,16,62,17,60,18,58,19,56,20,54,21,52,22,50,23,48,97,99,
%U A065263 101,103,105,107,109,111,113,115,117,119,121,123,125,127,32,126,33,124
%N A065263 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz1, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11AB..Y0 (where 1AB..Y0 is the complement of 0ab..y1).
%C A065263 When an infinite planar binary tree is mapped breadth-first-wise from left to right (1 is at top, 2 is its left, and 3 its right child, 4 is 2's left child, etc.) then this permutation induces such rearrangement of its nodes, that on the right side every node replaces its right child, on the left side the left children replace their parents, and the right children are reflected to the right side, to be the left children of their new parents.
%H A065263 Index entries for sequences that are permutations of the natural numbers
%p A065263 RightChildInverted := proc(n) local k; if(1 = n) then RETURN(3); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN((2*n)+1); fi; if(0 = (n mod 2)) then RETURN(n/2); fi; RETURN(2^(k+1) + ((2^(k+2))-1) - n); end;
%Y A065263 A057114, A065269, A065275, A065281, A065287. Inverse: A065264, conjugated with A059893: A065265, and the inverse of that: A065266.
%K A065263 nonn
%O A065263 1,1
%A A065263 Antti.Karttunen@iki.fi Oct 28 2001
%I A057114
%S A057114 3,1,7,2,6,14,15,4,5,12,13,28,29,30,31,8,9,10,11,24,25,26,27,56,57,58,59,60,61,62,63,16,17,
%T A057114 18,19,20,21,22,23,48,49,50,51,52,53,54,55,112,113,114,115,116,117,118,119,120,121,122,123,
%U A057114 124,125,126,127,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,96,97,98,99,100,101,102,103
%N A057114 Permutation of N induced by the order-preserving permutation of the rational numbers (x -> x+1); positions in Stern-Brocot tree.
%C A057114 The "unbalancing operation" used here is what is usually called "rotation of binary trees" (e.g. in Lucas, Ruskey et al. article)
%H A057114 A. Bogomolny, About the Stern-Brocot tree
%H A057114 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A057114 Joan Lucas, Dominique Roelants van Baronaigien and Frank Ruskey, On Rotations and the Generation of Binary Trees, Journal of Algorithms, 15 (1993) 343-366.
%H A057114 Index entries for sequences that are permutations of the natural numbers
%F A057114 a(n) = frac2position_in_whole_SB_tree(sbtree_perm_1_1_right(SternBrocotTreeNum(n)/SternBrocotTreeDen(n)))
%e A057114 Consider the following "extended" Stern-Brocot tree (on interval ]-inf,inf[):
%e A057114 ....................................0/1
%e A057114 .................-1/1.................................1/1
%e A057114 ......-2/1................-1/2...............1/2...............2/1
%e A057114 .-3/1......-3/2......-2/3......-1/3.....1/3.......2/3.....3/2.......3/1
%e A057114 Enumerate the fractions breadth-first (0/1 = 1, -1/1 = 2, 1/1 = 3, -2/1 = 4, -1/2 = 5, etc) then use this sequence to pick 3th, 1st, 7th, 2nd, etc fractions. We get a bijection (0/1 -> 1/1, -1/1 -> 0/1, 1/1 -> 2/1, -2/1 -> -1/1, -1/2 -> 1/2, etc) which is the function x -> x+1.
%e A057114 In other words, we cut the edge between 1/1 and 1/2, make 1/1 the new root, and create a new edge between 0/1 and 1/2 to get an "unbalanced" Stern-Brocot tree. If we instead make a similar change to subtree 1/1 (cut {2/1,3/2}, create {1/1,3/2} and make 2/1 the new root of the positive side, leaving the negative side as it is), we get the function given in Maple procedure sbtree_perm_1_1_right.
%e A057114 Both mappings belong to Cameron's group "A" of permutations of the rational numbers which preserve their linear order, and by applying such unbalancing operations successively (possibly infinitely many times) to the "extended" Stern-Brocot tree given above, the whole group "A" can be generated.
%p A057114 sbtree_perm_1_1_right := x -> (`if`((x <= 0),x,(`if`((x < (1/2)),(x/(1-x)),(`if`((x < 1),(3-(1/x)),(x+1)))))));
%Y A057114 SternBrocotNum given in A007305, SternBrocotDen in A047679, frac2position_in_whole_SB_tree in A054424. Inverse permutation: A057115. Cf. also A065249 and A065250.
%Y A057114 A065259(n) = A059893(A057114(A059893(n)))
%Y A057114 The first row of A065625, i.e. a(n) = RotateNodeRight(1, n).
%K A057114 nonn
%O A057114 1,1
%A A057114 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Aug 09 2000
%I A065259
%S A065259 3,1,7,2,11,5,15,4,19,9,23,6,27,13,31,8,35,17,39,10,43,21,47,12,51,25,
%T A065259 55,14,59,29,63,16,67,33,71,18,75,37,79,20,83,41,87,22,91,45,95,24,99,
%U A065259 49,103,26,107,53,111,28,115,57,119,30,123,61,127,32,131,65,135,34,139
%N A065259 A057114 conjugated with A059893, inverse of A065260.
%H A065259 Index entries for sequences that are permutations of the natural numbers
%F A065259 a(n) = A059893(A057114(A059893(n)))
%K A065259 nonn
%O A065259 1,1
%A A065259 Antti.Karttunen@iki.fi Oct 28 2001
%I A065289
%S A065289 3,1,7,2,13,5,15,4,25,6,27,9,29,11,31,8,49,10,51,12,53,14,55,17,57,19,
%T A065289 59,21,61,23,63,16,97,18,99,20,101,22,103,24,105,26,107,28,109,30,111,
%U A065289 33,113,35,115,37,117,39,119,41,121,43,123,45,125,47,127,32,193,34,195
%N A065289 A065287 conjugated with A059893, inverse of A065290.
%H A065289 Index entries for sequences that are permutations of the natural numbers
%F A065289 a(n) = A059893(A065287(A059893(n)))
%K A065289 nonn
%O A065289 1,1
%A A065289 Antti.Karttunen@iki.fi Oct 28 2001
%I A065265
%S A065265 3,1,7,2,13,5,15,4,25,6,27,11,29,9,31,8,49,10,51,12,53,14,55,23,57,21,
%T A065265 59,19,61,17,63,16,97,18,99,20,101,22,103,24,105,26,107,28,109,30,111,
%U A065265 47,113,45,115,43,117,41,119,39,121,37,123,35,125,33,127,32,193,34,195
%N A065265 A065263 conjugated with A059893, inverse of A065266.
%H A065265 Index entries for sequences that are permutations of the natural numbers
%F A065265 a(n) = A059893(A065263(A059893(n)))
%K A065265 nonn
%O A065265 1,1
%A A065265 Antti.Karttunen@iki.fi Oct 28 2001
%I A059090
%S A059090 1,1,1,1,3,1,7,3,1,1,15,30,30,5,1,31,195,605,780,543,300,135,45,10,1,1,
%T A059090 63,1050,9030,41545,118629,233821,329205,327915,224280,100716,29337,
%U A059090 5950,910,105,7
%N A059090 Triangle T(n,m) of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,..,A037952(n).
%C A059090 An antichain is called intersecting (or proper) antichain if every two members have a nonempty intersection. Row sums give the number of intersecting antichains on a labeled n-set or n-variable Boolean functions in the Post class F(7,2) or self-dual monotone Boolean functions of n+1 variables. Cf. A001206.
%D A059090 Jovovic V., Kilibarda G., The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
%D A059090 Pogosyan G., Miyakawa M., Nozaki A., Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.
%H A059090 Index entries for sequences related to Boolean functions
%H A059090 Pogosyan et al., The Number of Clique Boolean Functions
%F A059090 T(n,0)=1, T(n,1)=2^n-1, T(n,2)=A032263(n), T(n,3)=A051303(n), T(n,4)=A051304(n), T(n,5)=A051305(n), T(n,6)=A051306(n), T(n,7)=A051307(n).
%e A059090 [1], [1, 1], [1, 3], [1, 7, 3, 1], [1, 15, 30, 30, 5], [1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1], [1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7].
%Y A059090 Cf. A001206, A032263, A051303-A051307, A036239, A051180-A051185, A016269, A047707, A051112-A051118, A000372.
%K A059090 hard,nonn
%O A059090 0,5
%A A059090 Vladeta Jovovic, Goran Kilibarda (vladeta@Eunet.yu), Dec 28 2000
%I A050227
%S A050227 1,3,1,7,3,1,15,8,3,1,31,19,8,3,1,63,43,20,8,3,1,127,94,47,20,8,3,1,
%T A050227 255,201,107,48,20,8,3,1,511,423,238,111,48,20,8,3,1,1023,880,520,251,
%U A050227 112,48,20,8,3,1,2047,1815,1121,558,255,112,48,20,8,3,1,4095,3719
%N A050227 Triangle of number of n-tosses having a run of r or more heads for a fair coin with r=1 to n across and n=1, 2, ... down.
%D A050227 Feller, W. An Introduction to Probability Theory and Its Application, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
%H A050227 E. W. Weisstein, Link to a section of The World of Mathematics.
%H A050227 E. W. Weisstein, Link to a section of The World of Mathematics.
%e A050227 1; 3,1; 7,3,1; 15,8,3,1; ...
%Y A050227 Cf. A008466.
%K A050227 nonn,nice,tabl
%O A050227 1,2
%A A050227 Eric W. Weisstein (eric@weisstein.com)
%I A053092
%S A053092 1,1,3,1,7,3,5,1,15,7,3,11,5,19,9,1,31,15,7,13,25,3,23,11,21,5,19,37,9,
%T A053092 35,17,1,63,31,15,29,7,27,53,13,25,49,3,47,23,45,11,43,21,41
%N A053092 2^A053087 kara n.
%K A053092 nonn
%O A053092 1,3
%A A053092 Robert Lozyniak (11@onna.com), Feb 27 2000
%I A013602
%S A013602 1,1,1,3,1,7,3,27,1,3,7,15,43,15,3,3,15,25,31,7,15,15,7,15,21,55,21,
%T A013602 159,81,69,33,135,13,9,33,25,15,37,15,7,13,9,3,27,7,133,25,129,61,7,
%U A013602 277,267,111,99,33,27,25,43,33,25,451,277,67,7,51,169,67,27,85,87
%N A013602 nextprime(4^n)-4^n.
%p A013602 seq(nextprime(4^i)-4^i,i=0..100);
%K A013602 nonn
%O A013602 0,4
%A A013602 James Kilfiger (mapdn@csv.warwick.ac.uk)
%I A021319
%S A021319 0,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,
%T A021319 1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,
%U A021319 6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3,1,7,4,6,0,3
%N A021319 Decimal expansion of 1/315.
%K A021319 nonn,cons
%O A021319 0,3
%A A021319 njas
%I A010603
%S A010603 3,1,7,4,8,0,2,1,0,3,9,3,6,3,9,8,9,4,9,5,0,3,4,1,1,2,7,8,5,4,4,6,1,
%T A010603 6,5,2,0,7,8,2,9,8,6,6,5,5,7,9,9,7,0,6,0,1,9,6,1,6,5,7,1,5,2,3,6,5,
%U A010603 0,4,3,3,0,1,1,2,4,8,4,3,8,3,4,6,5,4,7,0,8,8,4,2,6,5,2,4,4,4,1,9,1
%N A010603 Decimal expansion of cube root of 32.
%K A010603 nonn,cons
%O A010603 1,1
%A A010603 njas
%I A021763
%S A021763 0,0,1,3,1,7,5,2,3,0,5,6,6,5,3,4,9,1,4,3,6,1,0,0,1,3,1,7,5,2,3,0,5,
%T A021763 6,6,5,3,4,9,1,4,3,6,1,0,0,1,3,1,7,5,2,3,0,5,6,6,5,3,4,9,1,4,3,6,1,
%U A021763 0,0,1,3,1,7,5,2,3,0,5,6,6,5,3,4,9,1,4,3,6,1,0,0,1,3,1,7,5,2,3,0,5
%N A021763 Decimal expansion of 1/759.
%K A021763 nonn,cons
%O A021763 0,4
%A A021763 njas
%I A054458
%S A054458 1,3,1,7,6,1,17,23,9,1,41,76,48,12,1,99,233,204,82,15,1,239,682,765,
%T A054458 428,125,18,1,577,1935,2649,1907,775,177,21,1,1393,5368,8680,7656,4010,
%U A054458 1272,238,24,1,3363,14641,27312,28548,18358,7506,1946,308,27,1,8119
%N A054458 Convolution triangle based on A001333(n), n >= 1.
%C A054458 In the language of the Shapiro et al. ref. (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
%C A054458 The G.f. for the row polynomials p(n,x) (increasing powers of x) is LPell(z)/(1-x*z*LPell(z)) with LPell(z) given in 'Formula'.
%C A054458 Column sequences are A001333(n+1), A054459(n), A054460(n) for m=0..2.
%F A054458 a(n,m):= ((n-m+1)*a(n,m-1) + (2n-m)*a(n-1,m-1) + (n-1)*a(n-2,m-1))/(4*m), n >= m >= 1; a(n,0)= A001333(n+1); a(n,m):= 0 if n= 0, with LPell(x)= (1+x)/(1-2*x-x^2)= G.f. of A001333(n+1).
%e A054458 {1}; {3,1}; {7,6,1}; {17,23,9,1};...
%e A054458 Fourth row polynomial (n=3): p(3,x)= 17+23*x+9*x^2+x^3
%Y A054458 Cf. A002203(n+1)/2. Row sums: A055099(n).
%K A054458 easy,nonn,tabl
%O A054458 0,2
%A A054458 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Apr 26 2000
%I A046913
%S A046913 1,3,1,7,6,3,8,15,1,18,12,7,14,24,6,31,18,3,20,42,8,36,24,15,31,
%T A046913 42,1,56,30,18,32,63,12,54,48,7,38,60,14,90,42,24,44,84,6,72,48,
%U A046913 31,57,93,18,98,54,3,72,120,20,90,60,42,62,96,8,127,84,36,68,126
%N A046913 Sum of divisors of n not congruent to 0 mod 3.
%e A046913 Divisors of 12 are 1 2 3 4 6 12, and discarding 3 6 and 12 we get a(12)=1+2+4=7.
%K A046913 nonn
%O A046913 1,2
%A A046913 njas
%I A058606
%S A058606 1,3,1,7,7,18,18,35,38,65,71,119,140
%N A058606 McKay-Thompson series of class 28A for Monster.
%D A058606 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058606 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058606 nonn
%O A058606 -1,2
%A A058606 njas, Nov 27, 2000
%I A016647
%S A016647 3,1,7,8,0,5,3,8,3,0,3,4,7,9,4,5,6,1,9,6,4,6,9,4,1,6,0,1,2,9,7,0,5,
%T A016647 5,4,0,8,8,7,3,9,9,0,9,6,0,9,0,3,5,1,5,2,1,4,0,9,6,7,3,4,3,6,2,1,1,
%U A016647 7,6,7,5,1,5,9,1,2,7,6,9,3,1,1,3,6,9,1,2,0,5,7,3,5,8,0,2,9,8,8,1,5
%N A016647 Decimal expansion of ln(24).
%D A016647 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
%K A016647 nonn,cons
%O A016647 1,1
%A A016647 njas
%I A019639
%S A019639 3,1,7,9,0,2,4,1,1,8,5,5,5,7,6,3,7,9,8,8,2,2,6,0,9,1,4,1,7,6,9,1,1,
%T A019639 7,4,2,2,6,5,0,4,0,1,4,7,4,6,0,8,6,9,8,8,0,8,8,8,9,7,3,1,7,3,3,6,9,
%U A019639 0,7,3,6,8,6,8,0,8,7,2,4,3,8,7,1,0,4,8,0,9,2,5,8,6,5,0,4,2,0,4,8,3
%N A019639 Decimal expansion of sqrt(2*Pi*E)/13.
%K A019639 nonn,cons
%O A019639 0,1
%A A019639 njas
%I A011207
%S A011207 1,1,3,1,7,9,2,7,9,1,1,5,2,7,8,6,6,3,2,6,0,5,7,8,2,2,7,7,6,4,2,9,7,
%T A011207 9,7,9,0,1,7,9,8,6,6,9,6,3,6,2,9,4,7,9,0,0,7,9,3,0,9,8,2,3,7,0,2,5,
%U A011207 1,3,6,1,9,7,5,4,1,8,7,4,4,2,8,5,4,5,8,6,5,4,7,2,0,8,6,9,1,7,0,3,3
%N A011207 Decimal expansion of 13th root of 5.
%K A011207 nonn,cons
%O A011207 1,3
%A A011207 njas
%I A011308
%S A011308 1,3,1,7,9,8,0,6,2,9,2,1,3,0,0,2,2,3,9,7,7,5,9,1,7,7,4,1,9,8,8,2,9,
%T A011308 6,2,7,7,0,8,2,9,7,6,7,3,9,9,6,1,3,7,4,6,4,3,5,9,2,4,6,6,7,2,9,8,6,
%U A011308 9,1,4,3,9,0,7,8,3,0,0,8,6,4,8,0,6,9,0,3,8,4,5,9,0,7,9,7,0,8,0,6,4
%N A011308 Decimal expansion of 9th root of 12.
%K A011308 nonn,cons
%O A011308 1,2
%A A011308 njas
%I A033465
%S A033465 1,3,1,7,9,11,13,3,17,19,21,23,1,27,29,31,33,7,37,39,41,
%T A033465 43,9,47,49,51,53,11,57,59,61,63,13,67,69,71,73,3,77,79,
%U A033465 81,83,17,87,89,91,93,19,97,99,101,103,21,107,109,111,113
%N A033465 First differences of sequence {1/(n^2+1)} (numerators).
%K A033465 nonn
%O A033465 0,2
%A A033465 njas
%I A051927
%S A051927 3,1,7,13,35,81,199,477,1155,2785,6727,16237,39203,94641,228487,551613,
%T A051927 1331715,3215041,7761799,18738637,45239075,109216785,263672647,
%U A051927 636562077,1536796803,3710155681,8957108167,21624372013,52205852195
%N A051927 Independent sets of vertices in graph K_2 X C_n (n > 2).
%F A051927 a(n) = a(n-1) + 3*a(n-2) + a(n-3)
%K A051927 easy,nonn
%O A051927 0,1
%A A051927 Stephen G. Penrice (spenrice@ets.org), Dec 19 1999
%I A036575
%S A036575 3,1,7,19,11,27,67,89,183,113,470,289,1142,970,1074,323,4827,1143,1346,
%T A036575 2921,9635,6797,4590,5423,23672,22073,8983,25359,91442,22851,22114,
%U A036575 18494,244362,69149,81310,106763,623731,667653,455730,241079,1093162
%N A036575 a(n) = least number not of form [ (a^2/n) ] + [ (b^2)/n ].
%K A036575 nonn
%O A036575 1,1
%A A036575 Dave Wilson
%I A060487
%S A060487 1,3,1,7,57,95,43,3,35,717,3107,4520,2465,445,12,155,7845,75835,244035,
%T A060487 325890,195215,50825,4710,70,651,81333,1653771,10418070,27074575,
%U A060487 33453959,20891962,6580070,965965,52430,465
%N A060487 Triangle T(n,k) of k-block tricoverings of an n-set (n >= 3, k >= 4).
%C A060487 A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.
%F A060487 E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k,3)*y).
%e A060487 [1, 3, 1], [7, 57, 95, 43, 3], [35, 717, 3107, 4520, 2465, 445, 12], [155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70], [651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465], .... There are 205 tricoverings of a 4-set(cf. A060486): 7 4-block, 57 5-block, 95 6-block, 43 7-block and 3 8-block tricoverings.
%Y A060487 Cf. A006095, A060483-A060485, (row sums) A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.
%K A060487 nonn
%O A060487 3,2
%A A060487 Vladeta Jovovic (vladeta@Eunet.yu), Mar 20 2001
%I A059526
%S A059526 0,3,1,8,1,3,1,5,0,5,2,0,4,7,6,4,1,3,5,3,1,2,6,5,4,2,5,1,5,8,7,6,6,4,5,
%T A059526 1,7,2,0,3,5,1,7,6,1,3,8,7,1,3,9,9,8,6,6,9,2,2,3,7,8,6,0,6,2,2,9,4,1,3,
%U A059526 8,7,1,5,5,7,6,2,6,9,7,9,2,3,2,4,8,6,3,8,4,8,9,8,6,3,6,1,6
%N A059526 Decimal expansion of the real part of ln(ln(ln(ln(ln(ln... ... ...(a+bi)))))...
%e A059526 Let R(x)=ln(ln(ln(ln...(x)))))))))))... Then R(x)= 0.31813150... + 1.33723570...i (except when x=0, x=1, x=e, x=e^e, x=e^(e^e) and so on.
%e A059526 R(x)=.31813150520476413531265425158766451720351761387139986692237860622941387155762697923248638489863616... + 1.3372357014306894089011621431937106125395021384605124188763127819143505313612049884188813234387940... *i.
%Y A059526 Cf. A059527.
%K A059526 cons,nonn,nice
%O A059526 1,2
%A A059526 Fabian Rothelius (fabian.rothelius@telia.com), Jan 21 2001
%E A059526 More terms from Vladeta Jovovic (vladeta@Eunet.yu), Feb 26 2001
%I A010288
%S A010288 3,1,8,2,1,8,2,11,1,6,8,2,10,2,1,2,2,3,2,4,3,10,3,13,1,
%T A010288 2,1,1,3,1,1,7,1,2,1,12,1,1,1,1,2,1,4,3,4,1,1,1,10,1,13,
%U A010288 3,5,1,5,1,1,1,1,5,4,4,1,1,4,2,3,6,1,3,30,1,11,3,1,10,8
%N A010288 Continued fraction for cube root of 59.
%K A010288 nonn,cofr
%O A010288 0,1
%A A010288 njas
%I A049541
%S A049541 3,1,8,3,0,9,8,8,6,1,8,3,7,9,0,6,7,1,5,3,7,7,6,7,5,2,6,7,4,5,0,2,8,
%T A049541 7,2,4,0,6,8,9,1,9,2,9,1,4,8,0,9,1,2,8,9,7,4,9,5,3,3,4,6,8,8,1,1,7,
%U A049541 7,9,3,5,9,5,2,6,8,4,5,3,0,7,0,1,8,0,2,2,7,6,0,5,5,3,2,5,0,6,1,7,1
%N A049541 Decimal expansion of 1/Pi.
%K A049541 nonn,cons
%O A049541 0,1
%A A049541 njas
%I A065451
%S A065451 0,1,1,1,1,3,1,8,3,8,3,55,3,144,8,21,21,987,8,2584,21,144,55,17711,21,
%T A065451 6765,144,2584,144,317811,21,832040,987,6765,987,46368,144,14930352,
%U A065451 2584,46368,987,102334155,144,267914296,6765,46368,17711,1836311903
%N A065451 Fibonacci(phi(n)).
%H A065451 Joseph L. Pe, The Euler Phibonacci Sequence: A Problem Proposal with Software
%e A065451 a(13) = F(phi(13)) = F(12) = 144.
%t A065451 Table[ Fibonacci[ EulerPhi[ n]], {n, 0, 60} ]
%o A065451 (PARI) for(n=1,75,print1(fibonacci(eulerphi(n)),","))
%Y A065451 A000045(A000010(n)). Cf. A065449.
%K A065451 nonn
%O A065451 0,6
%A A065451 Joseph L. Pe (joseph_l_pe@hotmail.com), Nov 18 2001
%E A065451 More terms from several correspondents, Nov 19, 2001
%I A054506
%S A054506 3,1,8,4,1,13,16,5,1,26,15,1,20,17,50,6,39,26,6,1,72,1,70,69,39,70,52,
%T A054506 1,1,72,1,41,87,81,82,101,94,27,108,56,116,84,181,1,43,1,46,208,1,74,
%U A054506 182,16,1,50,109,117,188,1,1,157,81,164,56,249,1,314,152,26,1,186,75
%N A054506 Log_b 3 where b is smallest primitive root (A001918) mod n-th prime.
%D A054506 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, Table 10.2, pp. 216-217.
%Y A054506 Cf. table in A054503. Also A054505-A054513.
%K A054506 nonn,easy
%O A054506 3,1
%A A054506 njas, Apr 09 2000
%E A054506 More terms from James A. Sellers (sellersj@math.psu.edu), Apr 09 2000
%I A055249
%S A055249 1,3,1,8,4,1,20,12,5,1,48,32,17,6,1,112,80,49,23,7,1,256,192,129,72,30,
%T A055249 8,1,576,448,321,201,102,38,9,1,1280,1024,769,522,303,140,47,10,1,2816,
%U A055249 2304,1793,1291,825,443,187,57,11,1,6144,5120,4097,3084,2116,1268,630
%N A055249 Triangle of partial row sums (prs) of triangle A055248 (prs of Pascal's triangle A007318).
%C A055249 In the language of the Shapiro et al. ref. (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z)/(1-2*z)^2)/(1-x*z/(1-z)).
%C A055249 This is the second member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
%C A055249 The column sequences appear in A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 for m=0..7.
%F A055249 a(n,m)=sum(A055248(n,k),k=m..n), n >= m >= 0, a(n,m):=0 if n= m >= 0, a(n,m):=0 if n= 0.
%e A055249 {1}; {3,1}; {8,4,1}; {20,12,5,1};...
%e A055249 Fourth row polynomial (n=3): p(3,x)= 20+12*x+5*x^2+x^3
%Y A055249 Cf. A007318, A055248, A008949. Row sums: A049611(n+1) = A055252(n, 0).
%K A055249 easy,nonn,tabl
%O A055249 0,2
%A A055249 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), May 26 2000
%I A021318
%S A021318 0,0,3,1,8,4,7,1,3,3,7,5,7,9,6,1,7,8,3,4,3,9,4,9,0,4,4,5,8,5,9,8,7,
%T A021318 2,6,1,1,4,6,4,9,6,8,1,5,2,8,6,6,2,4,2,0,3,8,2,1,6,5,6,0,5,0,9,5,5,
%U A021318 4,1,4,0,1,2,7,3,8,8,5,3,5,0,3,1,8,4,7,1,3,3,7,5,7,9,6,1,7,8,3,4,3
%N A021318 Decimal expansion of 1/314.
%K A021318 nonn,cons
%O A021318 0,3
%A A021318 njas
%I A016477
%S A016477 3,1,8,4,9,1,23,1,3,20,1,1,16,1,2,2,5,4,1,1,4,1,2,2,1,3,
%T A016477 1,1,1,1,2,1,5,5,1,1,3,1,11,1,2,3,2,1,3,4,1,1,1,11,2,1,
%U A016477 3,2,1,1,1,2,4,1,1,3,2,1,9,1,1,5,1,1,1,5,4,3,6,4,1,2,2
%N A016477 Continued fraction for ln(49).
%K A016477 nonn,cofr
%O A016477 1,1
%A A016477 njas
%I A062196
%S A062196 1,1,3,1,8,6,1,15,30,10,1,24,90,80,15,1,35,210,350,175,21,1,48,420,
%T A062196 1120,1050,336,28,1,63,756,2940,4410,2646,588,36,1,80,1260,6720,14700,
%U A062196 14112,5880,960,45,1,99,1980,13860
%N A062196 Coefficient triangle of certain polynomials N(2;m,x).
%C A062196 The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=2) Laguerre triangle L(2;n+m,m)= A062139(n+m,m), n >= 0, is N(2;m,x)/(1-x)^(3+2*m), with the row polynomials N(2;m,x):=sum(a(m,k)*x^k,k=0..m).
%F A062196 a(m,k)= [x^k]N(2;m,x), with N(2;m,x)= ((1-x)^(3+2*m))*diff((x^m)/(m!*(1-x)^(m+3)),x$m).
%F A062196 N(2;m,x)= sum((binomial(m,j)*(2*m+2-j)!/((m+2)!*(m-j)!)*(x^(m-j)))*(1-x)^j,j=0..m).
%Y A062196 A062145, A062190.
%K A062196 nonn,tabl
%O A062196 0,3
%A A062196 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Jun 19 2001
%I A030523
%S A030523 1,3,1,8,6,1,20,25,9,1,48,88,51,12,1,112,280,231,86,15,1,256,832,912,
%T A030523 476,130,18,1,576,2352,3276,2241,850,183,21,1,1280,6400,10976,9424,
%U A030523 4645,1380,245,24,1,2816,16896,34848,36432,22363,8583,2093,316,27,1
%N A030523 A convolution triangle of numbers obtained from A001792.
%C A030523 a(n,m):= s1p(3;n,m), a member of a sequence of unsigned triangles including s1p(2;n,m)= A007318(n-1,m-1) (Pascal's triangle). Signed version: (-1)^(n-m)*a(n,m):= s1(3;n,m).
%H A030523 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A030523 a(n,m) = 2*(2*m+n-1)*a(n-1,m)/n + m*a(n-1,m-1)/n, n >= m >= 1; a(n,m):=0, n= m >= 2.
%Y A039692 A039647, A000032, A000045.
%K A039692 nonn,tabl
%O A039692 1,2
%A A039692 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A049760
%S A049760 0,0,1,1,3,1,8,10,11,12,13,13,31,37,31,41,42,47,58,60,82,86,95,76,125,
%T A049760 123,140,103,115,134,188,229,235,213,186,239,264,283,244,243,263,342,
%U A049760 369,430,387,407,473,413,446,489,522,492,558
%N A049760 a(n)=SUM{T(n,k): k=1,2,...,n}, array T as in A049759.
%K A049760 nonn
%O A049760 1,5
%A A049760 Clark Kimberling, ck6@cedar.evansville.edu
%I A019146
%S A019146 0,3,1,8,11,25,38,77,180,434,1022,2460,6163,13595,34346,85405,
%T A019146 201291,534842
%N A019146 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite MEI = ZSM-18 Nan [ AlnSi34-nO68 ] . 28 H2O (n=2.1-5.7).
%D A019146 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019146 G. Thimm, Cycle sequences of crystal structures
%K A019146 nonn
%O A019146 3,2
%A A019146 Georg Thimm (mgeorg@ntu.edu.sg)
%I A049965
%S A049965 1,3,1,8,14,35,63,128,254,635,1205,2382,4743,9480,18953,37908,75814,
%T A049965 189535,360115,710757,1416777,2831193,5661209,11321848,22643315,
%U A049965 45286504,90572943,181145858,362291695,724583384,1449166761
%N A049965 a(n)=a(1)+a(2)+...+a(n-1)+a(m), where m=2^(p+1)+2-n, and 2^pEfficient Exhasutive Enumeration of one dimensional reversible cellular automata, 2001.
%e A063467 a(n)=1 for prime values of n as we must factor to get nontrivial examples.
%K A063467 nice,nonn
%O A063467 1,4
%A A063467 Tim Boykett (tim@timesup.org), Jul 27 2001
%I A021762
%S A021762 0,0,1,3,1,9,2,6,1,2,1,3,7,2,0,3,1,6,6,2,2,6,9,1,2,9,2,8,7,5,9,8,9,
%T A021762 4,4,5,9,1,0,2,9,0,2,3,7,4,6,7,0,1,8,4,6,9,6,5,6,9,9,2,0,8,4,4,3,2,
%U A021762 7,1,7,6,7,8,1,0,0,2,6,3,8,5,2,2,4,2,7,4,4,0,6,3,3,2,4,5,3,8,2,5,8
%N A021762 Decimal expansion of 1/758.
%K A021762 nonn,cons
%O A021762 0,4
%A A021762 njas
%I A019736
%S A019736 1,3,1,9,2,7,8,0,3,9,2,7,9,4,7,3,9,4,8,6,3,9,8,7,6,4,6,5,6,9,0,0,2,
%T A019736 3,8,1,7,3,7,2,0,9,8,2,8,4,5,3,1,5,4,6,4,8,2,4,3,6,8,0,1,8,8,2,2,8,
%U A019736 5,4,1,7,7,1,2,9,5,1,4,9,5,7,7,6,2,8,7,6,8,2,0,2,0,1,9,8,9,7,7,5,0
%N A019736 Decimal expansion of sqrt(2*Pi)/19.
%K A019736 nonn,cons
%O A019736 0,2
%A A019736 njas
%I A054448
%S A054448 1,3,1,9,4,1,26,14,5,1,73,44,20,6,1,201,131,69,27,7,1,545,376,220,102,
%T A054448 35,8,1,1460,1052,665,349,144,44,9,1,3873,2888,1937,1116,528,196,54,10,
%U A054448 1,10191,7813,5490,3402,1788,768,259,65,11,1,26633,20892,15240,10008
%N A054448 Triangle of partial row sums of triangle A054446(n,m), n >= m >= 0.
%C A054448 In the language of the Shapiro et al. ref. (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((Pell(z))^2)/(Fib(z)*(1-x*z*Fib(z))) with Pell(x)=1/(1-2*x-x^2) G.f. A000129(n+1) (Pell numbers without 0) and Fib(x)=1/(1-x-x^2) G.f. A000045(n+1) (Fibonacci numbers without 0).
%C A054448 This is the second member of the family of Riordan-type matrices obtained from the Fibonacci convolution matrix A037027 by repeated application of the partial row sums procedure.
%F A054448 a(n,m)=sum(A054446(n,k),k=m..n), n >= m >= 0, a(n,m):=0 if n= m >= 0, a(n,m):=0 if n= 0, with Fib(x)= G.f. A000045(n+1) and Pell(x)= G.f. A000129(n+1).
%e A054448 {1}; {3,1}; {9,4,1}; {26,14,5,1};...
%e A054448 Fourth row polynomial (n=3): p(3,x)= 26+14*x+5*x^2+x^3
%Y A054448 Cf. A037027, A000045, A000129. Row sums: A054449.
%K A054448 easy,nonn,tabl
%O A054448 0,2
%A A054448 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Apr 27 2000 and May 08 2000.
%I A016577
%S A016577 3,1,9,4,1,31,1,1,5,7,4,2,6,2,1,5,14,2,4,1,1,3,1,22,2,3,
%T A016577 2,2,2,1,12,2,1,1,1,1,1,2,6,2,2,4,1,7,2,1,9,27,2,1,14,6,
%U A016577 1,4,6,2,48,2,7,17,10,1,1,4,2,2,2,2,5,7,1,19,1,1,1,2,9
%N A016577 Continued fraction for ln(99/2).
%K A016577 nonn,cofr
%O A016577 1,1
%A A016577 njas
%I A021317
%S A021317 0,0,3,1,9,4,8,8,8,1,7,8,9,1,3,7,3,8,0,1,9,1,6,9,3,2,9,0,7,3,4,8,2,
%T A021317 4,2,8,1,1,5,0,1,5,9,7,4,4,4,0,8,9,4,5,6,8,6,9,0,0,9,5,8,4,6,6,4,5,
%U A021317 3,6,7,4,1,2,1,4,0,5,7,5,0,7,9,8,7,2,2,0,4,4,7,2,8,4,3,4,5,0,4,7,9
%N A021317 Decimal expansion of 1/313.
%K A021317 nonn,cons
%O A021317 0,3
%A A021317 njas
%I A005533 M2231
%S A005533 1,3,1,9,5,0,7,9,1,0,7,7,2,8,9,4,2,5,9,3,7,4,0,0,1,9,7,1,2,2,9,6,4,0,1,
%T A005533 3,3,0,3,3,4,6,9,0,1,3,1,9,3,4,1,8,6,8,1,5,0,5,8,0,7,7,9,5,9,8,0,5,3,5,9,8,0,8,9,3,5
%N A005533 Decimal expansion of fifth root of 4.
%K A005533 cons,nonn
%O A005533 1,2
%A A005533 njas
%I A050155
%S A050155 1,3,1,9,5,1,28,20,7,1,90,75,35,9,1,297,275,154,54,11,1,1001,1001,637,
%T A050155 273,77,13,1,3432,3640,2548,1260,440,104,15,1,11934,13260,9996,5508,
%U A050155 2244,663,135,17,1
%N A050155 T(n,k)=M(2n+2,n+2,k+2), 0<=k<=n, n >= 0, array M as in A050144.
%e A050155 Rows: {1}; {3,1}; {9,5,1}; ...
%K A050155 nonn,tabl
%O A050155 0,2
%A A050155 Clark Kimberling, ck6@cedar.evansville.edu
%I A027465
%S A027465 1,3,1,9,6,1,27,27,9,1,81,108,54,12,1,243,405,270,90,15,1,729,1458,
%T A027465 1215,540,135,18,1,2187,5103,5103,2835,945,189,21,1,6561,17496,20412,
%U A027465 13608,5670,1512,252,24,1,19683,59049,78732,61236,30618,10206,2268
%N A027465 Square of lower triangular normalized binomial matrix.
%D A027465 B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct. 1996), pp. 109-121.
%F A027465 Numerators of lower triangle of (b^2)[ i,j ] where b[ i,j ] = binomial(i-1,j-1)/2^(i-1) if j<=i, 0 if j>i.
%F A027465 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j).
%F A027465 a(n,m)= 4^(n-1)*sum(b(n,j)*b(j,m),j=m..n)= 3^(n-m)*binomial(n-1,m-1), n >= m >= 1; a(n,m):=0, nEncyclopedia of Combinatorial Structures 917
%F A052931 G.f.: -1/(-1+3*x^2+x^3)
%F A052931 Recurrence: {a(1)=0,a(0)=1,a(2)=3,a(n)+3*a(n+1)-a(n+3)}
%F A052931 Sum(1/9*(-1+5*_alpha+2*_alpha^2)*_alpha^(-1-n),_alpha=RootOf(-1+3*_Z^2+_Z^3))
%p A052931 spec:= [S,{S=Sequence(Prod(Z,Union(Z,Z,Z,Prod(Z,Z))))},unlabelled]: seq(combstruct[count](spec,size=n),n=0..20);
%K A052931 easy,nonn
%O A052931 0,3
%A A052931 encyclopedia@pommard.inria.fr, Jan 25 2000
%E A052931 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 06 2000
%I A006803 M2232
%S A006803 1,0,0,1,0,3,1,9,6,29,27,99,112,351,450,1275,1782,4704,6998,
%T A006803 17531,27324,65758,106211,247669,411291,935107,1587391,3535398,
%U A006803 6108103,13373929,23438144,50592067,89703467,191306745,342473589
%V A006803 1,0,0,-1,0,-3,1,-9,6,-29,27,-99,112,-351,450,-1275,1782,-4704,6998,
%W A006803 -17531,27324,-65758,106211,-247669,411291,-935107,1587391,-3535398,
%X A006803 6108103,-13373929,23438144,-50592067,89703467,-191306745,342473589
%N A006803 Percolation series for hexagonal lattice.
%C A006803 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%D A006803 J. W. Essam, A. J. Guttmann and K. De'Bell, On two-dimensional directed percolation, J. Phys. A 21 (1988), 3815-3832.
%D A006803 J. Blease, Series expansions for the directed-bond percolation problem, J. Phys. C 10 (1977), 917-924.
%D A006803 Jensen, Iwan; Guttmann, Anthony J.; Series expansions of the percolation probability for directed square and honeycomb lattices. J. Phys. A 28 (1995), no. 17, 4813-4833.
%H A006803 G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
%H A006803 I. Jensen, More terms
%Y A006803 Cf. A006809.
%K A006803 sign,done
%O A006803 0,6
%A A006803 njas
%I A019770
%S A019770 3,1,9,7,9,7,8,6,2,1,7,1,6,5,2,3,8,0,6,3,0,6,2,2,0,5,5,4,5,3,2,5,4,
%T A019770 4,1,1,5,0,0,8,5,2,5,9,9,2,5,8,8,1,8,7,7,3,5,2,5,5,2,5,6,0,3,2,6,1,
%U A019770 6,5,6,0,7,4,1,5,9,2,4,0,8,9,3,4,7,8,9,8,6,1,3,8,6,5,0,0,1,9,5,7,9
%N A019770 Decimal expansion of 2*E/17.
%K A019770 nonn,cons
%O A019770 0,1
%A A019770 njas
%I A016601
%S A016601 3,1,9,8,6,7,3,1,1,7,5,5,0,6,8,1,3,0,0,7,9,3,4,7,3,3,6,5,4,2,8,1,8,
%T A016601 2,8,9,1,1,9,8,6,6,9,3,2,4,8,0,3,4,6,7,1,2,2,7,9,8,1,0,0,2,9,0,3,8,
%U A016601 1,7,6,6,1,0,3,5,3,4,4,4,3,8,1,9,9,6,9,4,5,3,6,7,0,1,7,9,3,2,4,3,6
%N A016601 Decimal expansion of ln(49/2).
%K A016601 nonn,cons
%O A016601 1,1
%A A016601 njas
%I A021973
%S A021973 0,0,1,0,3,1,9,9,1,7,4,4,0,6,6,0,4,7,4,7,1,6,2,0,2,2,7,0,3,8,1,8,3,
%T A021973 6,9,4,5,3,0,4,4,3,7,5,6,4,4,9,9,4,8,4,0,0,4,1,2,7,9,6,6,9,7,6,2,6,
%U A021973 4,1,8,9,8,8,6,4,8,0,9,0,8,1,5,2,7,3,4,7,7,8,1,2,1,7,7,5,0,2,5,7,9
%N A021973 Decimal expansion of 1/969.
%K A021973 nonn,cons
%O A021973 0,5
%A A021973 njas
%I A056843
%S A056843 0,3,1,9,15,59,152,513,1539,4993,15836
%N A056843 Polydudes.
%H A056843 M. Vicher, Polyforms
%H A056843 More information
%K A056843 uned,nonn
%O A056843 0,2
%A A056843 James A. Sellers (sellersj@math.psu.edu), Aug 28 2000
%I A038202
%S A038202 1,1,3,1,9,27,15,18,288,288,420,464,1856,10080,46848,210240,400320,
%T A038202 652848,3991680,27528402,32659200,163296000,1143463200,1305467240,
%U A038202 6840489600,9453465438
%N A038202 Least k such that n!+k^2 is square.
%H A038202 E. W. Weisstein, Link to a section of The World of Mathematics.
%K A038202 nonn
%O A038202 4,3
%A A038202 dww
%I A010289
%S A010289 3,1,10,1,2,1,16,1,2,6,1,1,3,1,22,2,5,2,11,22,1,9,4,3,2,
%T A010289 2,4,7,1,2,1,1,2,1,1,2,1,12,2,3,1,5,2,1,1,4,2,214,1,3,24,
%U A010289 1,1,1,7,1,13,6,1,9,2,2,8,1,4,1,4,2,1,50,9,1,2,1,3,1,1
%N A010289 Continued fraction for cube root of 60.
%K A010289 nonn,cofr
%O A010289 0,1
%A A010289 njas
%I A019427
%S A019427 0,3,1,10,1,18,1,26,1,34,1,42,1,50,1,58,1,66,1,74,1,82,1,90,1,98,1,106,1,
%T A019427 114,1,122,1,130,1,138,1,146,1,154,1,162,1,170,1,178,1,186,1,194,1,202,1,
%U A019427 210,1,218,1,226,1,234,1,242,1,250,1,258,1,266,1,274,1,282,1,290,1,298,1
%N A019427 Continued fraction for tan(1/4).
%K A019427 nonn,cofr
%O A019427 0,2
%A A019427 dww
%I A008299
%S A008299 0,1,1,1,3,1,10,1,25,15,1,56,105,1,119,490,105,1,246,1918,1260,
%T A008299 1,501,6825,9450,945,1,1012,22935,56980,17325,1,2035,74316,
%U A008299 302995,190575,10395,1,4082
%N A008299 Triangle of associated Stirling numbers of second kind.
%C A008299 Rows are of lengths 1,1,2,2,3,3,...
%D A008299 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
%D A008299 A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.
%D A008299 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
%F A008299 S_r(n+1,k)=k S_r(n,k)+binomial(n,r-1)S_r(n-r+1,k-1) for this sequence, r=2 GF: sum(S_r(n,k)u^k ((t^n)/(n!)),n=0..infty,k=0..infty)=exp(u(e^t-sum(t^i/i!,i=0..r-1)))
%e A008299 There are 3 ways of partitioning a set N of cardinality 4 into 2 blocks each of cardinality at least 2, so S_2(4,2)=3.
%Y A008299 Rows give A000247, A000478, A058844. Cf. A059022, A059023, A059024, A059025.
%K A008299 nonn,tabf,nice
%O A008299 1,5
%A A008299 njas
%E A008299 Formula and cross-references from Barbara Haas Margolius (margolius@math.csuohio.edu), Dec 14 2000
%I A016478
%S A016478 3,1,10,2,1,2,1,2,18,1,1,6,1,1,1,7,1,9,1,9,3,1,4,1,1,3,
%T A016478 9,5,1,1,2,1,3,5,7,1,2,1,1,2,18,1,2,2,2,1,1,2,2,1,1,1,13,
%U A016478 5,515,8,4,2,2,1,1,1,44,4,6,8,4,1,5,2,3,51,2,11,2,1,7,2
%N A016478 Continued fraction for ln(50).
%K A016478 nonn,cofr
%O A016478 1,1
%A A016478 njas
%I A057967
%S A057967 1,3,1,10,5,2,30,21,11,3,83,75,49,18,5,208,231,177,84,30,6,495,636,554,
%T A057967 318,143,42,9,1101,1603,1540,1023,543,210,62,11,2327,3737,3907,2904,
%U A057967 1759,822,311,82,15,4685,8163,9153,7470,5012,2706,1219,423,111,18,9041
%N A057967 Triangle T(n,k) of numbers of minimal 4-covers of an unlabeled n+4-set that cover k points of that set uniquely (k=4,..,n+4).
%C A057967 Row sums give A005784.
%H A057967 More informations
%F A057967 T(n,k)=b(n,k)-b(n-1,k); b(n,k)=is coefficient of x^k in x^4/24*(Z(S_n;12+4*x,12+4*x^2,…)+8*Z(S_n;3+x,3+x^2,12+4*x^3,3+x^4,3+x^5,12+4*x^6,…)+6*Z(S_n;6+2*x,12+4*x^2,6+2*x^3,12+4*x^4,…)+3*Z(S_n;4,12+4*x^2,4,12+4*x^4,…)+6*Z(S_n;2,4,2,12+4*x^4,2,4,2,12+4*x^8,…)), where Z(S_n;x_1,x_2,...,x_n) is cycle index of symmetric group S_n of degree n.
%e A057967 [1], [3, 1], [10, 5, 2], [30, 21, 11, 3], [83, 75, 49, 18], ...,; there are 5 minimal 4-covers of an unlabeled 6-set that cover 5 points of that set uniquely.
%Y A057967 Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965(labeled case), A057966, A057968.
%K A057967 nonn,tabl
%O A057967 0,2
%A A057967 Vladeta Jovovic (vladeta@Eunet.yu), Oct 17 2000
%I A035324
%S A035324 1,3,1,10,6,1,35,29,9,1,126,130,57,12,1,462,562,312,94,15,1,1716,2380,
%T A035324 1578,608,140,18,1,6435,9949,7599,3525,1045,195,21,1,24310,41226,35401,
%U A035324 19044,6835,1650,259,24,1,92378,169766,161052,97954,40963,12021,2450
%N A035324 A convolution triangle of numbers, generalizing Pascal's triangle A007318.
%C A035324 Replacing each '2' in the recurrence by '1' produces Pascal's triangle A007318(n-1,m-1). The columns appear as A001700, A008549, A045720, A045894,...
%H A035324 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A035324 a(n+1,m) = 2*(2*n+m)*a(n,m)/(n+1) + m*a(n,m-1)/(n+1), n >= m >= 1; a(n,m):=0, n= 1.
%Y A035324 If offset 0 (n >= m >= 0): convolution triangle based on A001700 (central binomial coeffs. of odd order).
%K A035324 easy,nice,nonn,tabl
%O A035324 1,2
%A A035324 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A046658
%S A046658 1,3,1,10,7,1,35,38,11,1,126,187,82,15,1,462,874,515,142,19,1,1716,
%T A046658 3958,2934,1083,218,23,1,6435,17548,15694,7266,1955,310,27,1,24310,
%U A046658 76627,80324,44758,15086,3195,418,31,1,92378,330818,397923
%N A046658 Triangle related to A001700 and A000302 (powers of 4).
%F A046658 a(n,m) = binomial(n,m-1)*(binomial(2*n,n)/binomial(2*(m-1),m-1) - 4^(n-m+1)*(m-1)/n)/2, n >= m >= 1. G.f. for column m: x*c(x)*((x/(1-4*x))^(m-1))/sqrt(1-4*x); c(x = G.f. for Catalan numbers (A000108).
%Y A046658 Column sequences for m=1..6: A001700, A000531, A029887, A045724, A045492, A045530. Row sums: A046885.
%K A046658 easy,nonn,tabl
%O A046658 1,2
%A A046658 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A052964
%S A052964 1,0,3,1,10,7,35,36,127,165,474,715,1807,3004,6995,12393,27370,50559,
%T A052964 107883,204820,427351,826045,1698458,3321891,6765175,13333932,26985675,
%U A052964 53457121,107746282,214146295,430470899,857417220,1720537327
%N A052964 A simple regular expression.
%H A052964 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1035
%F A052964 G.f.: -(-1+x)/(1-x-3*x^2+2*x^3)
%F A052964 Recurrence: {a(1)=0,a(0)=1,a(2)=3,2*a(n)-3*a(n+1)-a(n+2)+a(n+3)}
%F A052964 Sum(-1/25*(-1-11*_alpha+6*_alpha^2)*_alpha^(-1-n),_alpha=RootOf(1-_Z-3*_Z^2+2*_Z^3))
%p A052964 spec:= [S,{S=Sequence(Prod(Union(Prod(Sequence(Z),Z),Z,Z),Z))},unlabelled ]: seq(combstruct[count ](spec,size=n),n=0..20);
%K A052964 easy,nonn
%O A052964 0,3
%A A052964 encyclopedia@pommard.inria.fr, Jan 25 2000
%E A052964 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 06 2000
%I A064060
%S A064060 1,1,1,3,1,10,15,1,10,15,15,45,60,90,1,21,35,70,105,105,105,105,210,
%T A064060 315,420,630,630,1,28,35,56,105,168,210,210,280,280,280,315,420,420,
%U A064060 560,560,840,840,840,1260,1260
%N A064060 Number of connected, homeomorphically irreducible (also called series reduced) trees with n >= 2 labeled leaves (numbers in non-decreasing order).
%C A064060 The number of entries of row n of this array is A007827(n), n >= 2.
%C A064060 With v the total number of nodes (vertices), e the number of edges (links), n >= 2 the number of edges ending in a degree 1 node (leaves), i the number of edges which end in nodes with degree >=3 (internal edges), and v_{d} the number of nodes of degree d=1,3,4,... one has: v= e+1 = n + sum(v_{d},d>=3), i= e-n, sum(d*v_{d},d>=3) = 2(v-1)-n.
%H A064060 Ch. Mayer, Illustration
%H A064060 Index entries for sequences related to trees
%e A064060 {1}; {1}; {1, 3}; {1, 10, 15}; {1, 10, 15, 15, 15, 45, 60, 90}; {1, 21, 35, 70, 105, 105, 105, 105, 210, 315, 420, 630, 630}; {1, 28, 35, 56, 105, 168, 210, 210, 280, 280, 280, 315, 420, 420, 560, 560, 840, 840, 840, 1260, 1260, 1680, 1680, 1680, 1680, 2520, 2520, 2520, 2520, 3360, 5040, 5040}, ...
%Y A064060 The row sums give A000311(n-1), n >= 2. Cf. A007827.
%K A064060 nonn,tabf
%O A064060 2,4
%A A064060 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Christoph Mayer (christoph.mayer@physik.uni-karlsruhe.de) Sep 13 2001
%I A048953
%S A048953 0,0,0,0,0,0,3,1,10,21
%N A048953 Non-trivial 3-component links with n crossings.
%D A048953 Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976.
%H A048953 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A048953 Cf. A048952.
%K A048953 nonn,more
%O A048953 0,7
%A A048953 Eric W. Weisstein (eric@weisstein.com)
%I A027446
%S A027446 1,3,1,11,5,2,25,13,7,3,137,77,47,27,12,147,87,57,37,22,10,1089,669,
%T A027446 459,319,214,130,60,2283,1443,1023,743,533,365,225,105,7129,4609,3349,
%U A027446 2509,1879,1375,955,595,280,7381,4861,3601,2761,2131,1627,1207,847
%N A027446 Square of the lower triangular mean matrix.
%F A027446 Numerators of lower triangle of (a[ i,j ])^2 where a[ i,j ] = 1/i if j<=i, 0 if j>i
%K A027446 nonn,tabl
%O A027446 1,2
%A A027446 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A027516
%S A027516 1,3,1,11,5,2,38,26,8,3,316,247,121,36,15,3183,2971,1508,558,180,70,
%T A027516 11265,11729,7032,2928,1178,364,154,14759,17212,10541,5127,2033,784,
%U A027516 259,105,17862,22396,15648,7593,3596,1379,567,182,77,22567,30830
%N A027516 Square of the lower triangular normalized partition matrix.
%F A027516 Numerators of lower triangle of (a[ i,j ])^2 where a[ i,j ] = T(i,j)/Partitions(i) if j<=i, 0 if j>i
%Y A027516 Cf. A008284.
%K A027516 nonn,tabl
%O A027516 1,2
%A A027516 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A008969
%S A008969 1,1,3,1,11,7,1,50,85,15,1,274,1660,575,31,1,1764,48076,46760,3661,63,
%T A008969 1,13068,1942416,6998824,1217776,22631,127,1,109584,104587344,
%U A008969 1744835904,929081776,30480800,137845,255
%N A008969 Triangle of differences of reciprocals of unity.
%D A008969 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
%K A008969 nonn,tabl
%O A008969 1,3
%A A008969 njas
%I A002185 M2233 N0885
%S A002185 3,1,11,43,19,683,2731,331,43691,174763,5419,2796203,251,87211,59,
%T A002185 715827883,67,281,1777,22366891,83,2932031007403,18837001,283,
%U A002185 4363953127297
%N A002185 Smallest primitive factor of 2^{2n+1} + 1.
%D A002185 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
%D A002185 M. Kraitchik, Recherches sur la Th\'{e}orie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.
%H A002185 S. S. Wagstaff, Jr., The Cunningham Project
%K A002185 nonn
%O A002185 0,1
%A A002185 njas
%I A002589 M2234 N0886
%S A002589 3,1,11,43,19,683,2731,331,43691,174763,5419,2796203,4051,87211,
%T A002589 3033169,715827883,20857,86171,25781083,22366891,8831418697,
%U A002589 2932031007403
%N A002589 Largest primitive factor of 2^{2n+1} + 1.
%D A002589 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
%D A002589 M. Kraitchik, Recherches sur la Th\'{e}orie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.
%H A002589 S. S. Wagstaff, Jr., The Cunningham Project
%K A002589 nonn
%O A002589 0,1
%A A002589 njas
%I A048522
%S A048522 1,3,1,11,57,51,17,187,953,947,913,827,313,2867,14609,13243,5049,46003,
%T A048522 234385,209723,69945,768819,3914001,3912635,3904441,3879859,3740561,
%U A048522 3388219,1282361,11746099,59848977,54211515,20517817,187937715
%N A048522 Terms of Binary Gleichniszahlen-Reihe (BGR) sequence A045998 converted into decimal.
%D A048522 N. Worrick, S. Lewis and B. Shrader, A possible formula for the length of BGR sequences, Graph Theory Notes of New York, XXXVI (1999), p. 25.
%Y A048522 Cf. A005150, A045998, A045999.
%K A048522 nonn,base
%O A048522 0,2
%A A048522 Patrick De Geest (pdg@worldofnumbers.com), Jun 1999.
%I A049458
%S A049458 1,3,1,12,7,1,60,47,12,1,360,342,119,18,1,2520,2754,1175,245,25,1,
%T A049458 20160,24552,12154,3135,445,33,1,181440,241128,133938,40369,7140,742,
%U A049458 42,1,1814400,2592720,1580508,537628
%V A049458 1,-3,1,12,-7,1,-60,47,-12,1,360,-342,119,-18,1,-2520,2754,-1175,
%W A049458 245,-25,1,20160,-24552,12154,-3135,445,-33,1,-181440,241128,
%X A049458 -133938,40369,-7140,742,-42,1,1814400,-2592720,1580508,-537628
%N A049458 Generalized Stirling number triangle of first kind.
%C A049458 a(n,m)= ^3P_n^m in the notation of the given ref. with a(0,0):=1. The monic row polynomials s(n,x):= sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(3+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
%C A049458 In the umbral calculus (see the S. Roman ref. given in A048854) the s(n,x) polynomials are called Sheffer polynomials for (exp(3*t),exp(t)-1).
%D A049458 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%F A049458 a(n,m)= a(n-1,m-1) - (n+2)*a(n-1,m), n >= m >= 0; a(n,m):=0,n= 1, and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman ref.).
%H A062139 Index entries for sequences related to Laguerre polynomials
%F A062139 a(n,m)=((-1)^m)*n!*binomial(n+2,n-m)/m!.
%F A062139 E.g.f. for m-th column seqence: ((-x/(1-x))^m)/(m!*(1-x)^3), m >= 0.
%e A062139 {1};{3,-1};{12,-8,1};{60,-60,15,-1};...; 2!*L(2,2,x)=12-8*x+x^2.
%Y A062139 For m=0..5 the (unsigned) columns give A001710, A005990, A005461, A062193-A062195. The row sums (signed) give A062197; the row sums (unsigned) give A062198.
%Y A062139 Cf. A021009, A062137-A062140, A066667.
%K A062139 sign,done,easy,tabl
%O A062139 0,2
%A A062139 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Jun 19 2001
%I A039811
%S A039811 1,3,1,12,9,1,60,75,18,1,358,660,255,30,1,2471,6288,3465,645,45,1,
%T A039811 19302,65051,47838,12495,1365,63,1,167894,728556,685580,235193,35700,
%U A039811 2562,84,1,1606137,8792910,10285488,4444188,877653,86940,4410,108,1
%N A039811 Matrix cube of Stirling-2 Triangle A008277.
%F A039811 E.g.f. k-th column: ((exp(exp(exp(x)-1)-1))^k)/k!.
%e A039811 1; 3,1; 12,9,1; 60,75,18,1; ...
%Y A039811 Cf. A039810-A039813. |a(n, 1)| = A000258(n) (first column).
%K A039811 nonn,tabl
%O A039811 1,2
%A A039811 Christian G. Bower (bowerc@usa.net), Feb 1999.
%I A046089
%S A046089 1,3,1,12,9,1,60,75,18,1,360,660,255,30,1,2520,6300,3465,645,45,1,
%T A046089 20160,65520,47880,12495,1365,63,1,181440,740880,687960,235305,35700,
%U A046089 2562,84,1,1814400,9072000,10372320,4452840,877905,86940,4410,108,1
%N A046089 A triangle of numbers related to triangle A030523.
%C A046089 a(n,1)= A001710(n+1). a(n,m)=: S1p(3;n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1;n,m)= A008275 (unsigned Stirling 1st kind), S1p(2;n,m)= A008297(n,m) (unsigned Lah numbers).
%C A046089 Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035342(n,m):=S2(3;n,m). The monic row polynomials E(n,x):=sum(a(n,m)*x^m,m=1..n), E(0,x):=1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%H A046089 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A046089 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A046089 a(n,m) = n!*A030523(n,m)/(m!*2^(n-m)); a(n,m) = (2*m+n-1)*a(n-1,m) + a(n-1,m-1), n >= m >= 1; a(n,m)=0, nCycle sequences of crystal structures
%K A019232 nonn
%O A019232 3,3
%A A019232 Georg Thimm (mgeorg@ntu.edu.sg)
%I A016479
%S A016479 3,1,13,1,2,68,1,59,4,9,3,1,1,3,1,17,1,2,4,4,1,4,1,1,2,
%T A016479 3,1,9,4,5,1,1,21,186,1,1,15,1,8,25,1,1,1,1,107,163,1,2,
%U A016479 1,1,2,9,10,2,2,4,1,1,121,1,2,19,1,2,1,47,1,2,2,1,3,4,6
%N A016479 Continued fraction for ln(51).
%K A016479 nonn,cofr
%O A016479 1,1
%A A016479 njas
%I A053286
%S A053286 1,1,3,1,13,17,45,1,189,225,685,257,2733,3841,12969,1,43693,75521,
%T A053286 174765,61697,731817,848897,2796205,262145,13304433,14802689,49449609,
%U A053286 15790337,185023425,313421825,715827885,6701057,2957312961,3551128577
%N A053286 Cototient of 2^n+1.
%F A053286 a(n)=A051593[A000051(n)]
%Y A053286 Cf. A000010, A000225, A051953.
%K A053286 nonn
%O A053286 1,3
%A A053286 Labos E. (labos@ana1.sote.hu), Mar 03 2000
%I A008826
%S A008826 1,1,3,1,13,18,1,50,205,180,1,201,1865,4245,2700,1,875,16674,
%T A008826 74165,114345,56700,1,4138,155477,1208830,3394790,3919860,
%U A008826 1587600
%N A008826 Triangle of coefficients from fractional iteration of e^x - 1.
%D A008826 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.
%e A008826 1; 1,3; 1,13,18; 1,50,205,180; ...
%Y A008826 Diagonals give A008827, A006472, A059355.
%K A008826 nonn,tabl,nice
%O A008826 2,3
%A A008826 njas
%I A010290
%S A010290 3,1,14,1,2,1,22,1,2,6,12,1,30,3,6,2,2,1,3,19,8,1,10,5,
%T A010290 4,3,2,6,30,2,17,9,1,5,1,1,9,4,3,6,4,1,6,3,18,1,1,3,2,1,
%U A010290 1,1,8,19,2,1,1,2,1,18,1,5,1,1,2,1,2,1,7,1,1,6,1,5,2,8
%N A010290 Continued fraction for cube root of 61.
%K A010290 nonn,cofr
%O A010290 0,1
%A A010290 njas
%I A018858
%S A018858 0,3,1,14,10,8,6,4,2,21,19,17,38,15,13,34,11,32,9,30,7,28,49,5,26,47,3,
%T A018858 24,45,66,22,43,108,20,85,41,106,18,83,39,104,60,16,81,37,102,14,123,79,
%U A018858 35,100,56,12,77,33,142,98,54,10,75,31,140,96,52,8,117,73,29,138,94,50,6
%N A018858 3^a(n) is smallest power of 3 beginning with n.
%K A018858 nonn,base
%O A018858 1,2
%A A018858 dww
%I A055301
%S A055301 1,1,0,1,0,3,1,14,20,77,308,147,3965,12527,26255,446923,1127033,
%T A055301 9020388,71170748,57286010,2168151928,17700260071,11899834419,
%U A055301 993097077555,7077421523510,12873010724109
%V A055301 1,-1,0,1,0,-3,-1,14,20,-77,-308,147,3965,12527,-26255,-446923,
%W A055301 -1127033,9020388,71170748,57286010,-2168151928,-17700260071,
%X A055301 -11899834419,993097077555,7077421523510,-12873010724109
%N A055301 Column 2 of triangle A055300.
%H A055301 Index entries for sequences related to trees
%K A055301 sign,done
%O A055301 2,6
%A A055301 Christian G. Bower (bowerc@usa.net), May 09 2000
%I A065250
%S A065250 3,1,15,5,12,7,63,2,23,48,6,29,60,31,255,9,20,11,95,192,24,51,26,14,
%T A065250 119,240,30,125,252,127,1023,4,39,80,10,45,92,47,383,768,96,195,98,25,
%U A065250 207,104,13,57,116,59,479,960,120,243,122,62,503,1008,126,509,1020,511
%N A065250 Permutation of N induced by the order-preserving permutation of the positive rational numbers (x -> 2x), positions in Stern-Brocot tree.
%H A065250 Index entries for sequences that are permutations of the natural numbers
%p A065250 [seq(A065250(j),j=1..120)]; A065250 := n -> frac2position_in_whole_SB_tree((SternBrocotTreeNum(n)/SternBrocotTreeDen(n))*2);
%Y A065250 Cf. A057114, A065251. Inverse permutation A065249.
%K A065250 nonn
%O A065250 1,1
%A A065250 Antti.Karttunen@iki.fi Oct 25 2001
%I A048966
%S A048966 1,3,1,15,6,1,90,39,9,1,594,270,72,12,1,4158,1953,567,114,15,1,30294,
%T A048966 14580,4482,1008,165,18,1,227205,111456,35721,8667,1620,225,21,1,
%U A048966 1741905,867834,287199,73656,15075,2430,294,24,1,13586859,6857136
%N A048966 A convolution triangle of numbers obtained from A025748.
%C A048966 A generalization of the Catalan triangle A033184.
%H A048966 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A048966 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A048966 a(n,m) = 3*(3*(n-1)-m)*a(n-1,m)/n + m*a(n-1,m-1)/n, n >= m >= 1; a(n,m):=0, n= 2. Row sums = A025756.
%K A048966 easy,nonn,tabl,nice
%O A048966 1,2
%A A048966 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A038553
%S A038553 1,1,3,1,15,6,7,1,63,30,341,12,819,14,15,1,255,126,9709,60,63,682,
%T A038553 2047,24,25575,1638,13797,28,475107,30,31,1
%N A038553 Max cycle length of differentiation digraph for n-bit binary sequences.
%D A038553 Simmons, G. J. The structure of the differentiation digraphs of binary sequences. Ars Combin. 35 (1993), A, 71-88. Math. Rev 95f:05052.
%K A038553 nonn
%O A038553 1,3
%A A038553 njas
%I A035342
%S A035342 1,3,1,15,9,1,105,87,18,1,945,975,285,30,1,10395,12645,4680,705,45,1,
%T A035342 135135,187425,82845,15960,1470,63,1,2027025,3133935,1595790,370125,
%U A035342 43890,2730,84,1,34459425,58437855,33453945,8998290
%N A035342 A triangle of numbers related to the triangle A035324; generalization of Stirling numbers of second kind A008277 and Lah numbers A008297.
%C A035342 If one replaces in the recurrence the '2' by '0', resp. '1', one obtains the Lah-number, resp. Stirling-number of 2nd kind, triangle A008297, resp. A008277.
%H A035342 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A035342 a(n,m) = n!*A035324(n,m)/(m!*2^(n-m)), n >= m >= 1; a(n+1,m)= (2*n+m)*a(n,m)+a(n,m-1); a(n,m):=0, n= 1.
%Y A035342 Cf. A000108, A035324, A008277, A008297.
%K A035342 easy,nice,nonn,tabl
%O A035342 1,2
%A A035342 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A039815
%S A039815 1,3,1,15,9,1,105,87,18,1,947,975,285,30,1,10472,12657,4680,
%T A039815 705,45,1,137337,188090,82887,15960,1470,63,1,2085605,3159699,
%U A039815 1598954,370237,43890,2730,84,1,36017472,59326371,33613353
%V A039815 1,-3,1,15,-9,1,-105,87,-18,1,947,-975,285,-30,1,-10472,12657,-4680,
%W A039815 705,-45,1,137337,-188090,82887,-15960,1470,-63,1,-2085605,3159699,
%X A039815 -1598954,370237,-43890,2730,-84,1,36017472,-59326371,33613353
%N A039815 Matrix cube of Stirling-1 Triangle A008275.
%F A039815 E.g.f. k-th column: ((ln(1+ln(1+ln(1+x))))^k)/k!.
%e A039815 1; -3,1; 15,-9,1; -105,87,-18,1; ...
%Y A039815 Cf. A039814-A039817. |a(n, 1)| = A000268(n) (first column).
%K A039815 sign,done,tabl
%O A039815 1,2
%A A039815 Christian G. Bower (bowerc@usa.net), Feb 1999.
%I A014621
%S A014621 1,1,3,1,15,10,3,1,105,105,55,30,10,3,1,945,1260,910,630,350,168,
%T A014621 76,30,10,3,1
%N A014621 Triangle of numbers arising from analysis of Levine's sequence A011784.
%K A014621 tabl,nonn
%O A014621 0,3
%A A014621 Colin Mallows colinm@research.avayalabs.com
%I A053485
%S A053485 1,1,1,3,1,15,45,21,315,2835,4725,22275,93555,2027025,42567525,
%T A053485 638512875,212837625,2170943775,13956067125,618718975875,9280784638125,
%U A053485 14992036723125,7522320180375,49308808782358125,147926426347074375
%N A053485 Denominators in expansion of exp(2x)/(1-x).
%Y A053485 Cf. A053484, A010842.
%K A053485 nonn,frac
%O A053485 0,4
%A A053485 njas, Jan 15 2000
%I A038675
%S A038675 1,1,3,1,16,10,1,55,165,35,1,156,1386,1456,126,1,399
%N A038675 Set x = n where x^n = SUM( A(n,k) * C( x+k-1, n) ), k=1.. and A(n,k) are the Eulerian numbers of A008292.
%C A038675 Andrews, Theory of Partitions, (1976), discussion of multisets.
%Y A038675 Cf. A001700, A014449, A000312.
%K A038675 nonn,tabl,more
%O A038675 0,3
%A A038675 Alford Arnold (Alford1940@aol.com)
%I A048159
%S A048159 1,3,1,16,13,3,125,171,85,15,1296,2551,2005,735,105,16807,43653,47586,
%T A048159 26950,7875,945,262144,850809,1195383,924238,412650,100485,10395,
%U A048159 4782969,18689527,32291463,31818045,19235755,7113645,1486485,135135
%N A048159 Triangle giving a(n,k) = number of (n,k) labeled Greg trees (n >= 2, 0<=k<=n-2).
%C A048159 An (n,k) Greg tree can be described as a tree with n black nodes and k white nodes where only the black nodes are labeled and the white nodes are of degree at least 3.
%D A048159 C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128.
%H A048159 Index entries for sequences related to trees
%F A048159 a(n,0)=n^(n-2), a(n,k)=(n+k-3)a(n-1,k-1)+(2n+2k-3)a(n-1,k)+(k+1)a(n-1,k+1).
%e A048159 1; 3,1; 16,13,3; 125,171,85,15;...
%Y A048159 Row sums give A005263. Cf. A005264, A048160, A052300-A052303.
%K A048159 nonn,easy,tabl,nice
%O A048159 2,2
%A A048159 njas
%E A048159 More terms from Larry Reeves (larryr@acm.org), Apr 07 2000
%I A060281
%S A060281 1,3,1,17,9,1,142,95,18,1,1569,1220,305,30,1,21576,18694,5595,745,45,1,
%T A060281 355081,334369,113974,18515,1540,63,1,6805296,6852460,2581964,484729,
%U A060281 49840,2842,84,1,148869153,158479488,64727522,13591116,1632099,116172
%N A060281 Triangle T(n,k) of labeled mappings from n points to themselves (endofunctions) with k cycles, k=1..n.
%D A060281 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
%F A060281 E.g.f.: 1/(1+LambertW(-x))^y.
%e A060281 [1], [3, 1], [17, 9, 1], [142, 95, 18, 1], [1569, 1220, 305, 30, 1], ... .
%Y A060281 Row sums: A000312.
%K A060281 easy,nonn,tabl
%O A060281 1,2
%A A060281 Vladeta Jovovic (vladeta@Eunet.yu), Apr 09 2001
%I A051141
%S A051141 1,3,1,18,9,1,162,99,18,1,1944,1350,315,30,1,29160,22194,6075,765,45,1,
%T A051141 524880,428652,131544,19845,1575,63,1,11022480,9526572,3191076,548289,
%U A051141 52920,2898,84,1,264539520,239660208
%V A051141 1,-3,1,18,-9,1,-162,99,-18,1,1944,-1350,315,-30,1,-29160,22194,
%W A051141 -6075,765,-45,1,524880,-428652,131544,-19845,1575,-63,1,-11022480,
%X A051141 9526572,-3191076,548289,-52920,2898,-84,1,264539520,-239660208
%N A051141 Generalized Stirling number triangle of first kind.
%C A051141 a(n,m)= R_n^m(a=0,b=3) in the notation of the given ref.
%C A051141 a(n,m) is a Jabotinsky matrix, i.e. the monic row polynomials E(n,x):=sum(a(n,m)*x^m,m=1..n) = product(x-3*j,j=0..n-1), n >= 1, E(0,x):=1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%D A051141 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%F A051141 a(n,m) = a(n-1,m-1) - 3*(n-1)*a(n-1,m), n >= m >= 1; a(n,m):=0,ni
%Y A027537 Cf. A008292.
%K A027537 nonn,tabl
%O A027537 1,2
%A A027537 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A002380 M2235 N0887
%S A002380 1,1,3,1,19,25,11,161,227,681,1019,3057,5075,15225,29291,55105,34243,
%T A002380 233801,439259,269201,1856179,3471385,6219851,1882337,5647011,50495465,
%U A002380 17268667,186023729,21200275,63600825,1264544299,3793632897,7085931395
%N A002380 3^n reduced modulo 2^n.
%D A002380 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 82.
%D A002380 S. S. Pillai, On Waring's problem, J. Indian Math. Soc., 2 (1936), 16-44.
%o A002380 (PARI.2.0.17) for(n=1,22,print(Mod(3^n,2^n)))
%K A002380 nonn,easy
%O A002380 1,3
%A A002380 njas
%E A002380 More terms from Jason Earls (jcearls@kskc.net), Jul 29 2001
%I A038455
%S A038455 1,3,1,20,9,1,210,107,18,1,3024,1650,335,30,1,55440,31594,7155,805,45,
%T A038455 1,1235520,725592,176554,22785,1645,63,1,32432400,19471500,4985316,
%U A038455 705649,59640,3010,84,1,980179200,598482000,159168428,24083892,2267769
%N A038455 A Jabotinsky-triangle related to A006963.
%C A038455 i) This triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z)= c(z) with c(z) ogf of Catalan numbers A000108. (Notation of F(z) as in Knuth's paper).
%C A038455 ii) E(n,x):= sum(a(n,m)*x^m,m=1..n), E(0,x)=1, are exponential convolution polynomials: E(n,x+y) = sum(binomial(n,k)*E(k,x)*E(n-k,y),k=0..n) (cf. Knuth's paper with E(n,x)= n!*F(n,x).)
%C A038455 iii) Explicit formula: see Knuth's paper for f(n,m) formula with f(k)= A006963(k+1).
%D A038455 D. E. Knuth, Convolution polynomials, The Mathematica J., 2.1 (1992) 67-78.
%F A038455 a(n,1) = A006963(n+1)=(2*n-1)!/n!, n >= 1; a(n,m) = sum(binomial(n-1,j-1)*A006963(j+1)*a(n-j,m-1),j=1..n-m+1), n >= m >= 2.
%Y A038455 A006963, A000108.
%K A038455 nonn,tabl
%O A038455 1,2
%A A038455 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A030042
%S A030042 1,1,3,1,20,24,63,288,1
%N A030042 Numerator of volume of best symplectic packing of n balls in 4-dimensional ball.
%D A030042 McDuff and Polterovich, Invent. Math. 115 (1994), 405-434.
%Y A030042 Cf. A030043.
%K A030042 nonn,frac
%O A030042 1,3
%A A030042 njas
%I A045496
%S A045496 1,1,3,1,20,24,63,288,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A045496 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A045496 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A045496 Packing density for n balls in complex projective plane using Fubini-Study metric (numerators).
%D A045496 D. McDuff, Notices Amer. Math. Soc. 45(8) (1998), pp. 952-960.
%Y A045496 Cf. A045496, A045497.
%K A045496 nonn,easy,frac
%O A045496 1,3
%A A045496 njas
%I A024432
%S A024432 1,0,3,1,20,41,147,1450,1441,27513,165814,166011,6759857,38623792,54257895,
%T A024432 2366944123,16391013794,27871290287,1112444405811,11151033695300,
%U A024432 24805899491473
%N A024432 a(n) = t(1) - t(2) + t(3) + ... + c*t(n), where c = (-1)^(n+1) and t(j) are Stirling numbers S(n,k) in decreasing order, for k = 1,2,...,n.
%K A024432 nonn
%O A024432 1,3
%A A024432 Clark Kimberling (ck6@cedar.evansville.edu)
%I A016531
%S A016531 1,3,1,21,1,6,1,2,1,2,13,1,1,1,1,2,2,16,5,1,2,5,1,1,9,3,
%T A016531 4,1,9,24,1,1,1,1,8,4,2,1,1,4,3,1,1,6,1,5,12,1,1,137,26,
%U A016531 1,2,2,2,1,7,2,9,1,7,1,1,2,60,2,1,2,1,9,6,1,1,6,3,2,1,54
%N A016531 Continued fraction for ln(7/2).
%K A016531 nonn,cofr
%O A016531 1,2
%A A016531 njas
%I A000369
%S A000369 1,3,1,21,9,1,231,111,18,1,3465,1785,345,30,1,65835,35595,7650,825,45,
%T A000369 1,1514205,848925,196245,24150,1680,63,1,40883535,23586255,5755050,
%U A000369 775845,62790,3066,84,1,1267389585,748471185,190482705,27478710
%N A000369 Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
%C A000369 a(n,m):= S2p(-3;n,m), a member of a sequence of triangles including S2p(-1;n,m):= A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1;n,m):= A008277(n,m) (Stirling 2nd kind). a(n,1)= A008545(n-1).
%H A000369 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A000369 Index entries for sequences related to Bessel functions or polynomials
%F A000369 a(n,m) = n!*A049213(n,m)/(m!*4^(n-m)); a(n+1,m) = (4*n-m)*a(n,m) + a(n,m-1), n >= m >= 1; a(n,m):=0, ni
%K A027477 nonn,tabl
%O A027477 1,2
%A A027477 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A033464
%S A033464 1,3,1,26,29,756,1793,45744,189513,4700260,30515629,730341600,
%T A033464 6948349069,159130156836,2123506814505,46081244842304,838034409016721,
%U A033464 17029766318842692,414549408916313189,7774211453384941440
%V A033464 1,3,-1,-26,29,756,-1793,-45744,189513,4700260,-30515629,-730341600,
%W A033464 6948349069,159130156836,-2123506814505,-46081244842304,838034409016721,
%X A033464 17029766318842692,-414549408916313189,-7774211453384941440
%N A033464 Logarithmic (or "LOG") transform of squares A000290.
%K A033464 sign,done
%O A033464 0,2
%A A033464 njas
%I A027495
%S A027495 1,3,1,27,21,2,173,227,48,2,2213,4283,1479,134,3,1239065,3303383,
%T A027495 1635807,238718,11475,156,1178661599,4156965497,2749126434,569141150,
%U A027495 43679400,1263600,11310,537440137205,2439821215787,2061668165820
%N A027495 Square of lower triangular normalized 2nd kind Stirling matrix.
%F A027495 Numerators of lower triangle of (a[ i,j ])^2 where a[ i,j ] = S(i,j)/BellNumber(i) if j<=i, 0 if j>i
%Y A027495 Cf. A008277.
%K A027495 nonn,tabl
%O A027495 1,2
%A A027495 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A046979
%S A046979 1,1,1,3,1,30,90,630,1,22680,113400,1247400,1,97297200,681080400,
%T A046979 10216206000,1,1389404016000,12504636144000,237588086736000,1,49893498214560000,
%U A046979 548828480360160000,12623055048283680000,1,3786916514485104000000
%N A046979 Denominators of Taylor series for exp(x)*sin(x).
%D A046979 G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%e A046979 1*x+1*x^2+1/3*x^3-1/30*x^5-1/90*x^6-1/630*x^7+1/22680*x^9+1/113400*x^10+...
%Y A046979 Cf. A046978.
%K A046979 nonn,frac,easy,nice
%O A046979 0,4
%A A046979 njas
%I A016481
%S A016481 3,1,32,1,1,1,18,2,97,2,1,1,1,28,2,2,1,331,6,1,1,1,7,2,
%T A016481 1,1,1,3,2,1,15,1,1,1,2,1,2,7,2,1,2,7,9,2,15,1,1,1,1,3,
%U A016481 1,26,6,1,1,2,2,6,25,1,1,14,31,1,5,11,4,2,1,2,2,1,2,6,3
%N A016481 Continued fraction for ln(53).
%K A016481 nonn,cofr
%O A016481 1,1
%A A016481 njas
%I A047815
%S A047815 1,1,3,1,33,5,3
%N A047815 Numerator of integral over [ 0,1 ] of n-th power of Cantor staircase function.
%e A047815 1/1 1/2 3/10 1/5 33/230 5/46 3/35
%Y A047815 Cf. A047816.
%K A047815 nonn,frac,more
%O A047815 0,3
%A A047815 Manfred Schroeder (Mrs17@aol.com)
%I A050817
%S A050817 3,1,41,5,9,265,35,89,7,93,23,8462643,383,27,95028841,97,169,39,937,
%T A050817 5105,8209,749,445,92307,81,64062862089,9862803,4825,3421,17067,9821,
%U A050817 480865,13,28230664709,3844609,5505,8223,17,25,359,4081,28481,11
%N A050817 Odd numbers seen in decimal expansion of pi (disregarding the decimal period) contiguous, smallest and distinct.
%C A050817 Leading zero not allowed thus forcing continuation of previous term.
%H A050817 C. Rivera, Problem 18 of C. Rivera's PP&P site.
%Y A050817 Cf. A047777, A050818, A016062, A050819, A050807.
%K A050817 nonn,easy,base
%O A050817 0,1
%A A050817 Patrick De Geest (pdg@worldofnumbers.com), Oct 1999.
%I A010292
%S A010292 3,1,46,1,2,1,70,1,2,5,3,4,1,2,94,1,8,4,1,6,1,1,1,7,3,3,
%T A010292 1,3,1,1,2,2,5,1,2,4,1,4,1,1,6,1,77,1,1,2,1,4,1,2,2,1,4,
%U A010292 1,2,64,1,1,9,6,10,1,6,1,1,2,1,1,3,1,1,2,53,1,2,24,2,1
%N A010292 Continued fraction for cube root of 63.
%K A010292 nonn,cofr
%O A010292 0,1
%A A010292 njas
%I A016482
%S A016482 3,1,89,1,3,2,35,1,1,1,1,39,3,1,1,1,14,4,13,2,6,1,3,1,1,
%T A016482 12,1,2,12,1,3,1,1,14,2,1,10,123,1,14,2,3,1,2,1,3,1,72,
%U A016482 2,8,21,1,1,1,5,1,25,4,4,1,1,89,3,1,2,1,1,4,3,6,1,3,5,59
%N A016482 Continued fraction for ln(54).
%K A016482 nonn,cofr
%O A016482 1,1
%A A016482 njas
%I A049330
%S A049330 1,1,3,1,115,11,5887,151,259723,15619,381773117,655177,20646903199,
%T A049330 27085381,467168310097,2330931341,75920439315929441,12157712239,
%U A049330 5278968781483042969,37307713155613,9093099984535515162569
%N A049330 Numerator of (1/pi)*Integral_{0..inf} (sin x / x)^n dx.
%D A049330 A. H. R. Grimsey, On the accumulation of chance effects and the Gaussian frequency distribution, Phil. Mag., 36 (1945), 294-295.
%D A049330 R. G. Medhurst and J. H. Roberts, Evaluation of the integral I_n(b) = of (2/pi)*Integral_{0..inf} (sin x / x)^n cos (bx) dx, Math. Comp., 19 (1965), 113-117.
%e A049330 1/2, 1/2, 3/8, 1/3, 115/384, 11/40, ...
%Y A049330 Cf. A049331. Same as A002297 except for n=4 term. Cf. also A002304, A002305.
%K A049330 nonn,frac,easy,nice
%O A049330 1,3
%A A049330 njas, Mark S. Riggs (msr1@ra.msstate.edu)
%I A036112
%S A036112 3,1,1311,1341,142351,1524231241,2534234251,3544336241,163554534221,
%T A036112 264554533231,266544534231,365554534221,366554434221,366554434221,
%U A036112 464564434221,463584334221,18362584434221,38262564535231
%N A036112 A summarize Fibonacci sequence: summarize the previous two terms!.
%C A036112 From the 82nd term the sequence gets into a cycle of 46.
%Y A036112 Cf. A036059.
%K A036112 base,easy,nonn
%O A036112 0,1
%A A036112 Floor van Lamoen (f.v.lamoen@wxs.nl)
%I A033909
%S A033909 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,1,0,0,
%T A033909 0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,3,2,0,0,0,0,
%U A033909 1,1,1,3,2,2,2,0,0,0
%N A033909 Number of Sort then Add steps needed to reach a sorted number, or -1 if never reach sorted number.
%e A033909 64 -> 64+46=110 -> 110+11=121 -> 121+112 = 233, taking 3 steps to reach a sorted number, so a(64)=3.
%Y A033909 Cf. A033863, A033908.
%K A033909 nonn,base
%O A033909 0,65
%A A033909 njas
%I A055654
%S A055654 0,0,0,1,0,0,0,3,2,0,0,3,0,0,0,7,0,4,0,5,0,0,0,9,4,0,8,7,0,0,0,15,0,0,
%T A055654 0,15,0,0,0,15,0,0,0,11,10,0,0,21,6,8,0,13,0,16,0,21,0,0,0,15,0,0,14,
%U A055654 31,0,0,0,17,0,0,0,37,0,0,12,19,0,0,0,35,26,0,0,21,0,0,0,33,0,20,0,23
%N A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.
%F A055654 a(n)=n-Sum[Phi[u]], where GCD[u, n/u] = 1, i.e. u is unitary divisor of n.
%e A055654 Square-free numbers are roots of a(n)=0 equation, while Min n for which a(n)=k is k^2. See also A000188,A008833.
%Y A055654 Cf. A000188, A008833, A055653, A053570, A053571, A055631, A055632.
%K A055654 nonn
%O A055654 1,8
%A A055654 Labos E. (labos@ana1.sote.hu), Jun 07 2000
%I A062787
%S A062787 3,2,0,0,4,2,6,6,10,7,15,9,18,18,21,15,28,18,31,29,34,24,41
%V A062787 -3,-2,0,0,4,2,6,6,10,7,15,9,18,18,21,15,28,18,31,29,34,24,41
%N A062787 [ exp(gamma) n log log n ] - phi(n), where gamma is Euler's constant (A001620).
%o A062787 (PARI.2.0.17) for(n=2,24,print(floor(exp(Euler)*n*log(log(n)))-eulerphi(n)))
%Y A062787 Cf. A058209.
%K A062787 easy,sign
%O A062787 2,1
%A A062787 Jason Earls (jcearls@kskc.net), Jul 18 2001
%I A062707
%S A062707 0,0,0,0,1,0,0,3,2,0,0,6,6,3,0,0,10,12,9,4,0,0,15,20,18,12,5,0,0,21,30,
%T A062707 30,24,15,6,0,0,28,42,45,40,30,18,7,0,0,36,56,63,60,50,36,21,8,0,0,45,
%U A062707 72,84,84,75,60,42,24,9,0,0,55,90,108,112,105,90,70,48,27,10,0,0,66
%N A062707 Table by anti-diagonals of nk(k+1)/2.
%F A062707 T(n,k) =T(n,1)*T(1,k) =A001477(n)*A000217(k) =A057145(n+2,k+1)-(k+1).
%e A062707 Rows start: 0 0 0 0 0 1 3 6 0 2 6 12 0 3 9 18
%Y A062707 Main diagonal is A002411. Sum of anti-diagonals is A000332.
%K A062707 easy,nonn,tabl
%O A062707 0,8
%A A062707 Henry Bottomley (se16@btinternet.com), Jul 11 2001
%I A059033
%S A059033 1,0,1,3,2,0,0,7,11,13,71,67,53,28,0,0,152,297,416,472,487,3965,
%T A059033 3890,3586,2921,2022,1015,0,0,8159,16300,23929,30243,34774,36804,
%U A059033 37306,398048,394008,377690,341125,289377,225082,152249,76140,0
%N A059033 Triangle in A059032 read by rows from left to right.
%Y A059033 Cf. A059032, A059034, A059035, A059037.
%K A059033 nonn,tabl
%O A059033 0,4
%A A059033 njas, Feb 13 2001
%I A008783
%S A008783 1,0,1,3,2,0,1,4,3,5,2,0,1,6,3,3,4,9,2,0,1,7,2,9,3,6,7,7,2,0,1,10,4,
%T A008783 7,4,3,5,8,5,10,2,0,1,12,5,3,4,15,3,14,4,12,4,16,2,0,1,9,2,19,2,16,6,
%U A008783 3,8,11,5,6,9,15,2,0,1,10,10,4,6,19,3,4,3,16,6,21,6,10,9,9,2,0,1,16,7
%N A008783 Period of c.f. representation of (sqrt(4n+1)+1)/2=sqrt(n+sqrt(n+sqrt(n+...))).
%K A008783 nonn,easy
%O A008783 1,4
%A A008783 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A054654
%S A054654 1,1,0,1,1,0,1,3,2,0,1,6,11,6,0,1,10,35,50,24,0,1,15,85,225,
%T A054654 274,120,0,1,21,175,735,1624,1764,720,0,1,28,322,1960,6769,
%U A054654 13132,13068,5040,0,1,36,546,4536,22449,67284,118124,109584
%V A054654 1,1,0,1,-1,0,1,-3,2,0,1,-6,11,-6,0,1,-10,35,-50,24,0,1,-15,85,-225,
%W A054654 274,-120,0,1,-21,175,-735,1624,-1764,720,0,1,-28,322,-1960,6769,
%X A054654 -13132,13068,-5040,0,1,-36,546,-4536,22449,-67284,118124,-109584
%N A054654 Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers of x.
%F A054654 n!*binomial(x,n)= Sum T(n,k)*x^(n-k), k=0..n.
%e A054654 3!*C(x,3) = x^3-3*x^2+2*x.
%e A054654 1; 1,0; 1,-1,0; 1,-3,2,0; ...
%o A054654 (PARI) T(n,k)=polcoeff(n!*binomial(x,n),n-k)
%Y A054654 Essentially Stirling numbers of 1st kind, multiplied by factorials - see A008276. Cf. A054655.
%K A054654 sign,done,easy,tabl,nice
%O A054654 0,8
%A A054654 njas, Apr 18 2000
%I A054503
%S A054503 0,0,1,0,1,3,2,0,2,1,4,5,3,0,1,8,2,4,9,7,3,6,5,0,1,4,2,9,5,11,3,8,10,7,
%T A054503 6,0,14,1,12,5,15,11,10,2,3,7,13,4,9,6,8,0,1,13,2,16,14,6,3,8,17,12,15,
%U A054503 5,7,11,4,10,9,0,2,16,4,1,18,19,6,10,3,9,20,14,21,17,8,7,12,15,5,13,11
%N A054503 Table T(n,k) giving log_b(k), 1<=k<=p, where p = n-th prime and b = smallest primitive root of p (A001918).
%D A054503 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, Table 10.2, pp. 216-217.
%e A054503 0; 0,1; 0,1,3,2; 0,2,1,4,5,3; 0,1,8,2,4,9,7,3,6,5; ...
%Y A054503 Columns of table give A054505-A054513.
%K A054503 nonn,tabf,nice,easy
%O A054503 0,6
%A A054503 njas, Apr 08 2000
%E A054503 More terms from James A. Sellers (sellersj@math.psu.edu), Apr 09 2000
%I A005874 M2236
%S A005874 0,3,2,0,3,12,0,6,0,6,0,12,6,6,12,12,3,0,2,6,0,24,0,24,6,3,0,24,6,12,
%T A005874 12,6,0,12,0,0,18,6,12,48,0,24,0,6,0,36,0,0,6,9,14,24,6,12,12,0,0,48,0,36,24,6,12
%N A005874 Theta series of hexagonal close-packing with respect to triangle between tetrahedra.
%D A005874 N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
%K A005874 nonn,easy
%O A005874 0,2
%A A005874 njas
%I A011231
%S A011231 1,3,2,0,4,6,9,2,4,7,7,5,6,1,2,3,7,9,1,8,0,9,3,2,7,3,3,1,5,0,0,2,6,
%T A011231 3,0,8,2,7,3,6,6,0,0,1,5,1,9,7,3,3,5,8,2,5,1,8,0,2,6,6,1,2,8,8,3,5,
%U A011231 4,6,7,1,7,4,3,6,5,2,8,9,1,9,8,5,5,3,3,7,2,8,8,2,4,0,9,4,6,7,3,2,4
%N A011231 Decimal expansion of 7th root of 7.
%K A011231 nonn,cons
%O A011231 1,2
%A A011231 njas
%I A021316
%S A021316 0,0,3,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,
%T A021316 2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,
%U A021316 1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8,2,0,5,1,2,8
%N A021316 Decimal expansion of 1/312.
%K A021316 nonn,cons
%O A021316 0,3
%A A021316 njas
%I A058096
%S A058096 1,0,3,2,0,6,5,0,3,6,0,18,12,0,21,16,0,6,27,0,60,34,0,72,51
%V A058096 1,0,-3,2,0,6,5,0,3,6,0,-18,12,0,21,16,0,6,27,0,-60,34,0,72,51
%N A058096 McKay-Thompson series of class 9d for Monster.
%D A058096 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058096 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058096 sign,done
%O A058096 -1,3
%A A058096 njas, Nov 27 2000
%I A049780
%S A049780 0,1,0,3,2,0,6,5,3,0,10,9,7,4,0,15,14,12,9,5,0,21,20,18,15,11,6,0,28,
%T A049780 27,25,22,18,13,7,0,36,35,33,30,26,21,15,8,0,45,44,42,39,35,30,24,17,9,
%U A049780 0,55,54,52,49,45,40,34,27,19,10,0,66,65,63
%N A049780 Array T by diagonals: T(m,n)=m+1 + m+2 + ... + m+n = (n/2)(n+2m+1).
%e A049780 Diagonals (each starting on row 1): {0}; {1,0}; {3,2,0}; ...
%Y A049780 Diagonal sums = A000330; see also A049777.
%K A049780 nonn,tabl
%O A049780 0,4
%A A049780 Clark Kimberling, ck6@cedar.evansville.edu
%I A010604
%S A010604 3,2,0,7,5,3,4,3,2,9,9,9,5,8,2,6,4,8,7,5,5,2,5,1,5,1,7,1,7,1,9,5,2,
%T A010604 0,1,1,1,3,6,1,8,5,1,6,6,3,3,6,0,5,7,2,1,7,1,7,2,4,0,7,1,8,4,7,2,8,
%U A010604 3,8,1,4,9,8,0,9,7,6,3,8,9,1,9,8,9,1,0,3,0,3,5,2,0,2,5,1,2,0,1,3,3
%N A010604 Decimal expansion of cube root of 33.
%K A010604 nonn,cons
%O A010604 1,1
%A A010604 njas
%I A048199
%S A048199 3,2,0,8,4,2,8,6,2,6,4,8,4,2,8,2,6,4,8,4,2,6,2,6,8,4,2,8,6,2,8,4,8,6,6,
%T A048199 4,8,2,8,2,6,4,4,2,8,6,4,2,8,6,2,6,4,4,8,2,6,4,8,4,2,2,8,4,2,8,4,8,8,6,
%U A048199 2,6,8,2,6,2,6,8,4,6,6,4,4,2,6,2,6,8,4,2,8,6,8,4,6,2,6,4,2,4,8,8,2,6,4
%N A048199 Distance of primes to next odd multiple of 5 (where n mod 10 = 5),
%e A048199 Take first prime 2, subtract from 5: 5-2=3, distance to 5; take 5-5 = 0; or 4th term, 15-7=8. Sequence may be converted to distance to nearest value of n (absolute value, where n mod 10 = 5) by changing all 8's to 2's and all 6's to 4's.
%Y A048199 A048198 and A007652.
%K A048199 easy,nonn
%O A048199 2,1
%A A048199 Enoch Haga (EnochHaga@msn.com )
%I A058257
%S A058257 1,0,1,0,0,1,1,1,1,0,3,2,1,0,0,0,3,5,6,6,6,0,0,3,8,14,20,26,71,71,71,
%T A058257 68,60,46,26,0,413,342,271,200,132,72,26,0,0,0,413,755,1026,1226,1358,
%U A058257 1430,1456,1456,1456,0,0,413,1168,2194,3420,4778,6208,7664,9120,10576
%N A058257 Triangle read by rows: this is a variant of A008280 in which 2 rows go from left to right, 2 from right to left, 2 from left to right, etc.
%C A058257 Suggested by first Atkinson reference.
%D A058257 M. D. Atkinson: Zigzag permutations and comparisons of adjacent elements, Information Processing Letters 21 (1985), 187-189.
%D A058257 M. D. Atkinson: Partial orders and comparison problems, Sixteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, February 1985), Congressus Numerantium 47, 77-88.
%H A058257 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
%H A058257 Index entries for sequences related to boustrophedon transform
%e A058257 1; 0,1; 0,0,1; 1,1,1,0; 3,2,1,0,0; 0,3,5,6,6,6; ...
%e A058257 1
%Y A058257 Cf. A058258, A008280, A000111.
%K A058257 nonn,easy,tabl,nice
%O A058257 0,11
%A A058257 njas, Dec 06 2000
%E A058257 More terms from Larry Reeves (larryr@acm.org), Dec 12 2000
%I A021761
%S A021761 0,0,1,3,2,1,0,0,3,9,6,3,0,1,1,8,8,9,0,3,5,6,6,7,1,0,7,0,0,1,3,2,1,
%T A021761 0,0,3,9,6,3,0,1,1,8,8,9,0,3,5,6,6,7,1,0,7,0,0,1,3,2,1,0,0,3,9,6,3,
%U A021761 0,1,1,8,8,9,0,3,5,6,6,7,1,0,7,0,0,1,3,2,1,0,0,3,9,6,3,0,1,1,8,8,9
%N A021761 Decimal expansion of 1/757.
%K A021761 nonn,cons
%O A021761 0,4
%A A021761 njas
%I A048797
%S A048797 1,0,0,1,0,1,0,3,2,1,0,1,0,3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0,1,0,5,4,3,2,
%T A048797 1,0,3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0,5,4,3,2,1,0,1,0,5,4,3,2,1,0,3,2,1,
%U A048797 0,1,0,5,4,3,2,1,0,3,2,1,0,5,4,3,2,1,0,7,6,5,4,3,2,1,0,3,2,1,0,1,0,3,2
%N A048797 Smallest number k such that n+k is prime.
%C A048797 Smarandache complementary prime function.
%D A048797 M. Popescu, V. Seleacu, About the Smarandache Complementary Prime Function, Smarandache Notions Journal, Vol. 7, No. 1-2-3, 1996, 12-22.
%H A048797 M. L. Perez et al., eds., Smarandache Notions Journal
%e A048797 a(1)=1 because 1+1=2=prime
%K A048797 nonn,easy,nice
%O A048797 0,8
%A A048797 Charles T. Le (charlestle@yahoo.com)
%E A048797 More terms from Joanna S. Bartlett (s1117611@cedarville.edu)
%I A007920
%S A007920 1,0,0,1,0,1,0,3,2,1,0,1,0,3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0,1,0,5,4,3,2,1,
%T A007920 0,3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0,5,4,3,2,1,0,1,0,5,4,3,2,1,0,3,2,1,0,1,
%U A007920 0,5,4,3,2,1,0,3,2,1,0,5,4,3,2,1,0,7,6,5,4,3,2,1,0,3,2,1,0,1,0,3,2,1,0,1
%N A007920 Smallest k such that n+k is prime.
%D A007920 F. Smarandache, "Only Problems, not Solutions!", Xiquan Publ., Phoenix-Chicago, 1993
%H A007920 M. L. Perez et al., eds., Smarandache Notions Journal
%K A007920 nonn
%O A007920 1,8
%A A007920 R. Muller
%I A056898
%S A056898 1,0,0,1,0,1,0,3,2,1,0,1,0,3,2,1,0,1,0,3,4,1,0,7,2,9,2,1,0,1,0,3,2,3,6,
%T A056898 1,0,3,2,1,0,1,0,3,4,1,0,5,2,3,4,1,0,5,2,9,2,1,0,1,0,3,2,3,6,1,0,9,2,1,
%U A056898 0,1,0,3,2,5,6,1,0,3,4,1,0,5,2,9,4,1,0,7,4,3,2,3,6,1,0,3,2
%N A056898 Smallest number where a(n)^2+n is prime.
%F A056898 a(n) =sqrt(A056896(n)-n) =sqrt(A056897(n)). For p a prime: a(p)=0 (and a(p-1)=1 if p<>3).
%e A056898 a(8)=3 since 3^2+8=17 which is prime
%Y A056898 Cf. A000040, A002496, A056892-A056898.
%K A056898 nonn
%O A056898 1,8
%A A056898 Henry Bottomley (se16@btinternet.com), Jul 05 2000
%I A062160
%S A062160 0,1,0,1,1,0,1,0,1,0,1,1,1,1,0,1,0,3,2,1,0,1,1,5,7,3,1,0,1,0,11,20,13,
%T A062160 4,1,0,1,1,21,61,51,21,5,1,0,1,0,43,182,205,104,31,6,1,0,1,1,85,547,
%U A062160 819,521,185,43,7,1,0,1,0,171,1640,3277,2604,1111,300,57,8,1,0,1,1,341
%V A062160 0,1,0,-1,1,0,1,0,1,0,-1,1,1,1,0,1,0,3,2,1,0,-1,1,5,7,3,1,0,1,0,11,20,13,4,1,0,-1,1,
%W A062160 21,61,51,21,5,1,0,1,0,43,182,205,104,31,6,1,0,-1,1,85,547,819,521,185,43,7,1,0,1,0,
%X A062160 171,1640,3277,2604,1111,300,57,8,1,0,-1,1,341,4921,13107,13021,6665,2101,455,73,9,1
%N A062160 Table by anti-diagonals of T(n,k)=(n^k-(-1)^k)/(n+1).
%F A062160 T(n,k) =n^(k-1)-n^(k-2)+n^(k-3)-...1 =n^(k-1)-T(n,k-1) =nT(n,k-1)-(-1)^n =(n-1)T(n,k-1)+nT(n,k-2) =round[n^k/(n+1)] for n>1.
%Y A062160 Rows include A062157, A000035, A001045, A015518, A015521, A015531, A015540, A015552, A015565, A015577, A015585, A015592 etc. Columns include A000004, A000012, A023443, A002061, A062158, A060884, A062159, A060888 etc. Related to repunits in negative bases (cf. A055129 for positive bases).
%K A062160 sign,tabl
%O A062160 0,18
%A A062160 Henry Bottomley (se16@btinternet.com), Jun 08 2001
%I A022959
%S A022959 3,2,1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,
%T A022959 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,
%U A022959 38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55
%V A022959 3,2,1,0,-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-11,-12,-13,-14,-15,-16,-17,-18,-19,
%W A022959 -20,-21,-22,-23,-24,-25,-26,-27,-28,-29,-30,-31,-32,-33,-34,-35,-36,-37,
%X A022959 -38,-39,-40,-41,-42,-43,-44,-45,-46,-47,-48,-49,-50,-51,-52,-53,-54,-55
%N A022959 3-n.
%K A022959 sign,done
%O A022959 0,1
%A A022959 njas
%I A023445
%S A023445 3,2,1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,
%T A023445 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,
%U A023445 38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55
%V A023445 -3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,
%W A023445 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,
%X A023445 38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55
%N A023445 n-3.
%K A023445 sign,done
%O A023445 0,1
%A A023445 njas
%I A031251
%S A031251 3,2,1,0,3,4,3,3,3,2,3,1,3,0,2,4,2,3,2,2,2,1,2,0,1,4,1,3,1,2,
%T A031251 1,1,1,0,0,4,0,3,0,2,0,1,0,0,3,4,4,3,4,3,3,4,2,3,4,1,3,4,0,3,
%U A031251 3,4,3,3,3,3,3,2,3,3,1,3,3,0,3,2,4,3,2,3,3,2,2,3,2,1,3,2,0,3
%N A031251 Write n in base 5, then complement each digit (d -> 4-d), and juxtapose.
%K A031251 nonn
%O A031251 1,1
%A A031251 Clark Kimberling, ck6@cedar.evansville.edu
%I A051427
%S A051427 0,0,0,0,0,0,0,3,2,1,0,6,1
%N A051427 Strictly Deza graphs with n nodes.
%D A051427 M. Erickson et al., Deza graphs: a generalization of strongly regular graphs, J. Comb. Des., 7 (1999), 395-405.
%K A051427 nonn,nice
%O A051427 1,8
%A A051427 njas
%I A035327
%S A035327 1,0,1,0,3,2,1,0,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,
%T A035327 1,0,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,
%U A035327 10,9,8,7,6,5,4,3,2,1,0,63,62
%N A035327 Write n in binary, interchange 0's and 1's.
%F A035327 a(n) = 2^( INT( Log2(n) ) + 1 ) - n - 1 - from Artemario Tadeu M da Silva (artemario@uol.com.br), Mar 19 2001
%e A035327 8 = 1000 -> 0111 = 111 = 7
%K A035327 nonn,easy,base
%O A035327 0,5
%A A035327 njas
%I A004444
%S A004444 3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12,19,18,17,16,23,22,
%T A004444 21,20,27,26,25,24,31,30,29,28,35,34,33,32,39,38,37,36,
%U A004444 43,42,41,40,47,46,45,44,51,50,49,48,55,54,53,52,59,58
%N A004444 Nimsum n + 3.
%D A004444 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
%D A004444 J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%H A004444 Index entries for sequences related to Nim-sums
%K A004444 nonn
%O A004444 0,1
%A A004444 njas
%I A035103
%S A035103 1,0,1,0,1,1,3,2,1,1,0,3,3,2,1,2,1,1,4,3,4,2,3,3,4,3,2,2,2,3,0,5,5,4,4,
%T A035103 3,3,4,3,3,3,3,1,5,4,3,3,1,3,3,3,1,3,1,7,5,5,4,5,5,4,5,4,3,4,3,4,5,3,3,
%U A035103 5,3,2,3,2,1,5,4,5,4,4,4,2,4,2,2,5,4,3,2,3,1,2,2,2,1,1,7,6,5,6,5,5,5,4
%N A035103 Number of 0's in binary representation of n-th prime.
%t A035103 Table[ Count[ IntegerDigits[ Prime[ n ],2 ],0 ],{n,120} ]
%Y A035103 Cf. A014499, A035100.
%K A035103 nonn,base,easy
%O A035103 1,7
%A A035103 njas
%E A035103 More terms from Erich Friedman (erich.friedman@stetson.edu)
%I A016557
%S A016557 3,2,1,1,1,1,25,1,2,48,1,1,1,1,1,1,18,1,60,1,2,44,1,1,8,
%T A016557 1,1,2,2,2,1,1,1,13,157,2,6,1,1,1,8,10,4,53,1,1,1,3,2,1,
%U A016557 2,10,6,2,1,2,1,1,1,15,1,2,2,13,2,8,2,2,1,1,2,10,1,5,1
%N A016557 Continued fraction for ln(59/2).
%K A016557 nonn,cofr
%O A016557 1,1
%A A016557 njas
%I A027082
%S A027082 1,1,1,1,1,1,1,3,2,1,1,1,3,5,6,5,1,1,1,3,5,9,14,16,11,1,1,1,3,
%T A027082 5,9,17,28,39,41,27,1,1,1,3,5,9,17,31,54,84,108,107,68,1,1,1,
%U A027082 3,5,9,17,31,57,102,169,246,299,283,175,1,1,1,3,5,9,17,31,57
%N A027082 Triangular array T by rows: T(n,0)=1 for n >= 0, T(n,1)=T(n,2)=0 for n >= 1, and for n >= 2, T(n,k)=T(n-1,k-3)+T(n-1,k-2)+T(n-1,k-1) for 3<=k<=2n-1, T(n,2n)=T(n-1,2n-3)+T(n-1,2n-2).
%K A027082 nonn,tabl
%O A027082 1,8
%A A027082 Clark Kimberling, ck6@cedar.evansville.edu
%I A010268
%S A010268 3,2,1,1,3,1,19,1,1,3,2,1,2,1,2,3,1,1,29,1,21,1,1,1,3,9,
%T A010268 2,2,3,2,4,4,1,5,4,1,39,1,7,1,6,1,173,3,4,7,1,1,6,2,3,1,
%U A010268 1,2,9,1,6,1,1,79,1,7,1,24,2,1,2,1,1,1,7,5,1,2,4,1,1,1
%N A010268 Continued fraction for cube root of 39.
%K A010268 nonn,cofr
%O A010268 0,1
%A A010268 njas
%I A053542
%S A053542 1,1,3,2,1,1,3,2,1,1,3,2,1,5,4,3,2,1,1,5,4,3,2,1,3,2,1,1,3,2,1,5,4,3,2,
%T A053542 1,5,4,3,2,1,1,5,4,3,2,1,3,2,1,1,5,4,3,2,1,3,2,1,5,4,3,2,1,7,6,5,4,3,2,
%U A053542 1,3,2,1,1,3,2,1,1,3,2,1,13,12,11,10,9,8,7,6,5,4,3,2,1,3,2,1,5,4,3,2,1
%N A053542 Distance to next prime in sequence of composites.
%F A053542 Beginning with 2, make sequence of composites; measure distances to next prime
%e A053542 a(1)=4, composite; distance to next prime, 5, is 1.
%K A053542 easy,nonn
%O A053542 1,3
%A A053542 Enoch Haga (EnochHaga@msn.com), Jan 16 2000
%I A053989
%S A053989 3,2,1,1,4,1,2,1,2,2,4,1,8,1,2,2,4,1,2,1,2,2,6,1,6,4,2,3,6,1,2,1,4,2,4,
%T A053989 2,2,1,6,2,4,1,6,1,2,3,6,1,2,3,2,2,4,1,2,3,2,3,6,1,8,1,4,2,6,2,6,1,2,2,
%U A053989 4,1,14,1,2,2,4,3,2,1,8,2,4,1,6,3,2,3,16,1,2,4,6,3,4,2,2,1,2,2
%N A053989 Smallest k such that nk-1 is prime.
%F A053989 a(n)=(A038700(n)+1)/n
%e A053989 a(5)=4 because the smallest prime in the sequence 5k-1 (4,9,14,19,24...) is 19 when k=4
%Y A053989 Cf. A034693, A038700.
%K A053989 easy,nonn
%O A053989 1,1
%A A053989 Henry Bottomley (se16@btinternet.com), Apr 04 2000
%I A046225
%S A046225 1,1,1,1,3,2,1,1,5,2,1,1,11,2,7,2,1,1,9,1,9,2,1,1,20,1,27,2,11,2,1,1,
%T A046225 67,2,19,1,13,2,1,1,147,2,105,2,51,2,15,2,1,1,126,1,78,1,33,1,17,2,1,1
%N A046225 First numerator and then denominator of elements to right of central elements of 1/2-Pascal triangle.
%e A046225 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...
%Y A046225 Cf. A046213.
%K A046225 tabl,nonn
%O A046225 1,5
%A A046225 Mohammad K. Azarian, ma3@cedar.evansville.edu
%I A058280
%S A058280 1,1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,4,1,2,4,1,288,1,90,1,12,1,1,
%T A058280 7,1,3,1,6,1,2,71,9,3,1,5,36,1,2,2,1,1,1,2,5,9,8,1,7,1,2,2,1,63,1,4,3,
%U A058280 1,6,1,1,1,5,1,9,2,5,4,1,2,1,1,2,20,1,1,2,1,10,5,2,1,100,11,1,9,1,2,1
%N A058280 Continued fraction for square root of Pi.
%H A058280 Index entries for continued fractions for constants
%t A058280 ContinuedFraction[ Sqrt[Pi], 100]
%K A058280 cofr,nonn,easy
%O A058280 0,3
%A A058280 Robert G. Wilson v (rgwv@kspaint.com), Dec 07 2000
%E A058280 More terms from Harvey P. Dale (hpd1@is2.nyu.edu), Dec 29 2000
%I A016457
%S A016457 3,2,1,2,1,1,1,1,7,1,1,6,3,1,1,1,4,2,6,2,1,1,5,2,1,1,2,
%T A016457 2,23,358,1,5,1,3,4,3,2,3,4,3,1,6,4,1,3,8,482,4,4,2,12,
%U A016457 2,1,17,49,7,2,1,6,1,3,2,1,1,3,1,9,1,145,41,2,480,4,10
%N A016457 Continued fraction for ln(29).
%K A016457 nonn,cofr
%O A016457 1,1
%A A016457 njas
%I A060567
%S A060567 3,2,1,2,2,2,4,2,2,6,5,4,10,4,2,8,10,6,12,4,6,18,10,4,14,14,6,10,23,20,
%T A060567 22,10,18,22,10,18,34,26,15,18,24,14,32,14,10,42,28,12,28,18,12,32,34,
%U A060567 14,25,26,42,54,30,14,58,42,22,24,28,36,60,38,20,34,44,26,70,42,20,42
%N A060567 Number of binomial coefficients C[n,j] with j=0...n that are divisible by C[n,2].
%F A060567 a(n)=Cardinality{j|Mod[C(n,j),C(n,2)]=0, j=0..n}; For n=1 Mod[1,0] does not exist.
%e A060567 The relevant residues are for n=16: {1, 16, 0, 80, 20, 48, 88, 40, 30, 40, 88, 48, 20, 80, 0, 16, 1}, a(16)=2; n=227: only with 6 non-zero residues: {1, 227, [111 zero], 454, 454, [111 zeros], 227, 1}
%Y A060567 A060430, A014070.
%K A060567 nonn
%O A060567 2,1
%A A060567 Labos E. (labos@ana1.sote.hu), Apr 12 2001
%I A036583
%S A036583 1,3,2,1,2,3,1,3,2,3,1,2,1,3,2,1,2,3,1,2,1,3,2,3,1,3,2,1,2,3,1,3,2,
%T A036583 3,1,2,1,3,2,3,1,3,2,1,2,3,1,2,1,3,2,1,2,3,1,3,2,3,1,2,1,3,2,1,2,3,
%U A036583 1,2,1,3,2,3,1,3,2,1,2,3,1,2,1,3,2,1,2,3,1,3,2,3,1,2,1,3,2,3,1,3,2
%N A036583 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.
%D A036583 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.
%K A036583 nonn
%O A036583 0,2
%A A036583 njas
%I A047878
%S A047878 0,3,2,1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,7,8,9,8,9,10,9,10,11,10,11,
%T A047878 12,11,12,13,12,13,14,13,14,15,14,15,16,15,16,17,16,17,18,17,18,19,18,
%U A047878 19,20,19,20,21,20,21,22,21,22,23,22,23,24,23,24,25
%N A047878 a(n)=least number of knight's moves from corner (0,0) to n-th diagonal of unbounded chessboard.
%C A047878 a(n)=MIN{T(i,n-i): i=0,1,...,n}, array T as in A049604.
%F A047878 a(3n)=n, a(3n+1)=n+1, a(3n+2)=n+2 for n >= 1.
%K A047878 nonn
%O A047878 0,2
%A A047878 Clark Kimberling, ck6@cedar.evansville.edu
%I A048984
%S A048984 1,1,1,3,2,1,2,4,3,1,2,6,4,1,12,5,10,5,1,6,11,7,15,3,8,14,6,2,8,12,10,
%T A048984 3,28,7,19,11,4,26,10,22,14,2,14,20,15,32,5,15,27,8,6,17,21,17,41,6,12,
%U A048984 33,20,4,43,21,35,19,3,50,22,38,24,50,10,19,37
%N A048984 As n runs through composite numbers, a(n) = number of nonprime d < n such that GCD(d,n) = 1.
%e A048984 9 is 4th composite number, GCD(9,1)=GCD(9,4)=GCD(9,8)=1, so a(4) = 3.
%Y A048984 Cf. A000010, A048983.
%K A048984 easy,nonn
%O A048984 0,4
%A A048984 Naohiro Nomoto (6284968128@geocities.co.jp)
%I A036585
%S A036585 3,2,1,3,1,2,3,2,1,2,3,1,3,2,1,3,1,2,3,1,3,2,1,2,3,2,1,3,1,2,3,2,1,
%T A036585 2,3,1,3,2,1,2,3,2,1,3,1,2,3,1,3,2,1,3,1,2,3,2,1,2,3,1,3,2,1,3,1,2,
%U A036585 3,1,3,2,1,2,3,2,1,3,1,2,3,1,3,2,1,3,1,2,3,2,1,2,3,1,3,2,1,2,3,2,1
%N A036585 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.
%D A036585 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.
%K A036585 nonn
%O A036585 0,1
%A A036585 njas
%I A010267
%S A010267 3,2,1,3,4,1,2,2,1,2,1,1,1,3,13,2,3,1,4,1,1,1,572,1,1,1,
%T A010267 1,10,1,1,2,1,9,1,1,1,1,1,1,2,1,1,74,4,2,1,11,2,1,2,1,2,
%U A010267 1,6,2,1,1,1,1,1,1,3,9,1,1,8,1,1,6,6,1,1,161,1,2,2,1,2
%N A010267 Continued fraction for cube root of 38.
%K A010267 nonn,cofr
%O A010267 0,1
%A A010267 njas
%I A023636
%S A023636 1,3,2,1,3,4,1,2,3,1,2,2,1,4,3,1,2,4,1,3,4,1,2,2,1,3,2,1,4,3,1,2,2,
%T A023636 1,4,2,1,3,2,1,2,4,1,3,4,1,2,3,1,4,2,1,2,3,1,2,4,1,3,4,1,2,2,1,3,4,
%U A023636 1,2,3,1,4,3,1,2,2,1,4,2,1,3,2,1,2,4,1,3,2,1,4,2,1,2,3,1,4,3,1,2,4
%N A023636 a(n) = s(3n) - s(3n-1), where s( ) is sequence A023635.
%K A023636 nonn
%O A023636 1,2
%A A023636 Clark Kimberling (ck6@cedar.evansville.edu)
%I A035572
%S A035572 1,0,0,1,1,0,1,1,3,2,1,3,5,6,4,5,12,11,14,13,17,29,28,30,40,46,68,67,75,
%T A035572 99,116,149,158,175,237,266,327,354,402,528,591,683,782,886,1127,1249,
%U A035572 1425,1654,1893,2310,2572,2902,3419,3879,4626,5158,5818,6837,7752,9027
%N A035572 Partitions of n with equal number of parts congruent to each of 0, 1 and 2 (mod 5)
%K A035572 nonn,part
%O A035572 0,9
%A A035572 Olivier Gerard (ogerard@ext.jussieu.fr)
%E A035572 More terms from dww
%I A025261
%S A025261 3,2,1,3,7,16,40,111,327,984,2984,9120,28156,87830,276442,876443,2795523,
%T A025261 8963668,28877692,93433104,303474772,989159058,3234373630,10606540108,
%U A025261 34874855416,114951540690,379751773694,1257173264286,4170000218994
%V A025261 3,-2,1,3,7,16,40,111,327,984,2984,9120,28156,87830,276442,876443,2795523,
%W A025261 8963668,28877692,93433104,303474772,989159058,3234373630,10606540108,
%X A025261 34874855416,114951540690,379751773694,1257173264286,4170000218994
%N A025261 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.
%K A025261 sign,done
%O A025261 1,1
%A A025261 Clark Kimberling (ck6@cedar.evansville.edu)
%I A046900
%S A046900 1,1,1,1,3,2,1,3,10,6,1,9,10,42,24,17,21,50,42,216,120
%V A046900 1,-1,1,1,-3,2,1,3,-10,6,-1,9,10,-42,24,-17,21,50,42,-216,120
%N A046900 Triangle inverse to that in A046899.
%C A046900 Sequence gives numerators; denominators are A001813.
%D A046900 H. W. Gould, A class of binomial sums and a series transformation, Utilitas Math., 45 (1994), 71-83.
%e A046900 1; -1/2 1/2; 1/12 -3/12 2/12; ...
%Y A046900 Cf. A046899.
%K A046900 sign,done,easy,nice,more
%O A046900 0,5
%A A046900 njas
%I A060848
%S A060848 1,3,2,1,4,2,5,4,3,2,6,2,3,4,8,1,4,4,6,9,12,6,4,12,6,7,30,4,12,12,5,16,
%T A060848 6,4,10,10,12,10,6,3,4,6,10,4,6,2,4,10,6,17,4,10,4,18,6,30,12,12,4,10,
%U A060848 27,4,6,4,12,4,28,6,2,10,4,4,10,12,18,10,10,3,12,4,12,6,10,10,18,10,12
%N A060848 Difference between a nontrivial prime power (A025475) and the next prime.
%F A060848 a(n)=nextprime[A025475(n)]-A025475(n)=A013632[A025475(n)]
%e A060848 78125=5^7 is followed by 78137, the difference is 12; a(n)=1 only for some powers of 2.
%Y A060848 Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711.
%K A060848 nonn
%O A060848 1,2
%A A060848 Labos E. (labos@ana1.sote.hu), May 03 2001
%I A006020 M2237
%S A006020 3,2,1,4,3,1,3,3,2,5,4,5,2,5,1,3,3,2,5,4,3,4,5,2,5,4,3,5,4,3,2,5,3,5,5,
%T A006020 3,4,5,2,5,4,2,5,4,7,2,5,2,5,4,2,4,5,3,5,4,5,2,5,4,2,4,4,7,2,1,4,3,3,5,5,2,3,4,5
%N A006020 Suspense numbers for Tribulations.
%D A006020 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 502.
%K A006020 nonn
%O A006020 1,1
%A A006020 njas
%I A028412
%S A028412 1,1,1,1,3,2,1,4,8,3,1,7,17,21,5,1,11,48,72,55,8,1,18,122,329,305,144,
%T A028412 13,1,29,323,1353,2255,1292,377,21,1,47,842,5796,15005,15456,5473,987,
%U A028412 34,1,76,2208,24447,104005,166408,105937,23184,2584,55,1,123,5777
%N A028412 Triangle of numbers Fibonacci(m(n+1))/Fibonacci(m).
%D A028412 I. Strazdins, Lucas factors and a Fibonomial generating function, in Applications of Fibonacci numbers, Vol. 7 (Graz, 1996), 401-404, Kluwer Acad. Publ., Dordrecht, 1998.
%e A028412 1; 1,1; 1,3,2; 1,4,8,3; ...
%K A028412 tabl,nonn,easy,nice
%O A028412 0,5
%A A028412 njas
%E A028412 More terms from Erich Friedman (efriedma@stetson.edu), Jun 03 2001
%I A030313
%S A030313 1,3,2,1,5,2,1,1,3,1,3,2,7,2,1,1,3,1,1,2,1,1,2,4,1,3,2,1,4,2,
%T A030313 4,3,9,2,1,1,3,1,1,2,1,1,2,4,1,1,2,1,1,1,1,3,1,1,2,2,2,1,3,5,
%U A030313 1,3,2,1,4,2,1,3,1,2,2,5,2,4,3,1,5,3,5,4,11,2,1,1,3,1,1,2,1,1
%N A030313 Length of n-th run of 1's in A030308.
%K A030313 nonn
%O A030313 1,2
%A A030313 Clark Kimberling, ck6@cedar.evansville.edu
%I A019587
%S A019587 1,1,3,2,1,5,3,8,5,2,9,5,1,10,5,15,9,3,15,8,21,13,5,20,11,2,19,9,
%T A019587 27,16,5,25,13,1,23,10,33,19,5,30,15,41,25,9,37,20,3,33,15,46,27,
%U A019587 8,41,21,55,34
%N A019587 The left budding sequence: # of i such that 0Classic Sequences
%Y A019587 Cf. A019588.
%K A019587 nonn,easy,nice
%O A019587 1,3
%A A019587 njas, jhc
%I A021315
%S A021315 0,0,3,2,1,5,4,3,4,0,8,3,6,0,1,2,8,6,1,7,3,6,3,3,4,4,0,5,1,4,4,6,9,
%T A021315 4,5,3,3,7,6,2,0,5,7,8,7,7,8,1,3,5,0,4,8,2,3,1,5,1,1,2,5,4,0,1,9,2,
%U A021315 9,2,6,0,4,5,0,1,6,0,7,7,1,7,0,4,1,8,0,0,6,4,3,0,8,6,8,1,6,7,2,0,2
%N A021315 Decimal expansion of 1/311.
%K A021315 nonn,cons
%O A021315 0,3
%A A021315 njas
%I A065366
%S A065366 1,3,2,1,5,6,7,3,4,5,1,2,3,11,10,9,13,12,11,15,14,13,7,6,5,9,8,7,11,10,
%T A065366 9,3,2,1,5,4,3,7,6,5,21,22,23,19,20,21,17,18,19,25,26,27,23,24,25,21,
%U A065366 22,23,29,30,31,27,28,29,25,26,27,13,14,15,11,12,13,9,10,11,17,18,19
%V A065366 1,-3,-2,-1,5,6,7,3,4,5,1,2,3,-11,-10,-9,-13,-12,-11,-15,-14,-13,-7,-6,-5,-9,-8,-7,
%W A065366 -11,-10,-9,-3,-2,-1,-5,-4,-3,-7,-6,-5,21,22,23,19,20,21,17,18,19,25,26,27,23,24,25,
%X A065366 21,22,23,29,30,31,27,28,29,25,26,27,13,14,15,11,12,13,9,10,11,17,18,19,15
%N A065366 Replace 3^k with (-2)^k in balanced ternary expansion of n.
%C A065366 Notation: (3)< n >(2)
%e A065366 5 = +1(9)-1(3)-1(1) --> +1(4)-1(-2)-1(1) = 5 = a(5)
%Y A065366 A065364, A065365
%K A065366 base,easy,sign,done
%O A065366 1,2
%A A065366 Marc Le Brun (mlb@well.com), Oct 31 2001
%I A021912
%S A021912 0,0,1,1,0,1,3,2,1,5,8,5,9,0,3,0,8,3,7,0,0,4,4,0,5,2,8,6,3,4,3,6,1,
%T A021912 2,3,3,4,8,0,1,7,6,2,1,1,4,5,3,7,4,4,4,9,3,3,9,2,0,7,0,4,8,4,5,8,1,
%U A021912 4,9,7,7,9,7,3,5,6,8,2,8,1,9,3,8,3,2,5,9,9,1,1,8,9,4,2,7,3,1,2,7,7
%N A021912 Decimal expansion of 1/908.
%K A021912 nonn,cons
%O A021912 0,7
%A A021912 njas
%I A050165
%S A050165 1,1,1,1,3,2,1,5,9,5,1,7,20,28,14,1,9,35,75,90,42,1,11,54,154,275,297,
%T A050165 132,1,13,77,273,637,1001,1001,429,1,15,104,440,1260,2548,3640,3432,
%U A050165 1430,1,17,135,663,2244,5508,9996,13260,11934
%N A050165 T(n,k)=M(2n+1,k,-1), 0<=k<=n, n >= 0, array M as in A050144.
%C A050165 T is a mirror image of the array in A039599.
%e A050165 Rows: {1}; {1,1}; {1,3,2}; ...
%K A050165 nonn,tabl
%O A050165 0,5
%A A050165 Clark Kimberling, ck6@cedar.evansville.edu
%I A033878
%S A033878 1,1,1,1,3,2,1,5,10,6,1,7,22,38,22
%N A033878 Triangular array associated with Schroeder numbers.
%H A033878 E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths, and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
%K A033878 nonn,tabl,more,easy
%O A033878 0,5
%A A033878 njas
%I A002130 M2238 N0888
%S A002130 1,1,1,3,2,1,5,23,25,27,49,74,62,85
%V A002130 1,-1,1,3,-2,1,-5,23,-25,27,-49,74,-62,85
%N A002130 Generalized sum of divisors function.
%D A002130 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
%Y A002130 A diagonal of A060044.
%K A002130 sign,done,easy,more
%O A002130 3,4
%A A002130 njas
%I A031252
%S A031252 3,2,1,6,3,31,22,12,25,23,2,19,10,15,13,11,1,7,5,36,26,16,6,46,
%T A031252 33,253,178,3,255,32,31,175,100,206,22,29,161,97,131,12,27,86,
%U A031252 71,56,207,25,166,91,195,192,23,163,88,180,2,30,21,85,165,162
%N A031252 Least k such that base 5 representation of n begins at s(k), where s=A031251.
%K A031252 nonn
%O A031252 1,1
%A A031252 Clark Kimberling, ck6@cedar.evansville.edu
%I A052174
%S A052174 1,1,1,3,2,1,6,8,3,1,20,20,15,4,1,50,75,45,24,5,1,175,210,189,84,35,
%T A052174 6,1,490,784,588,392,140,48,7,1,1764,2352,2352,1344,720,216,63,8,1,
%U A052174 5292,8820,7560,5760,2700,1215,315,80,9,1
%N A052174 Triangle of numbers arising in enumeration of walks on square lattice.
%H A052174 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
%e A052174 1; 1 1; 3 2 1; 6 8 3 1; 20 20 15 4 1; 50 75 45 24 5 1; 175 210 189 84 35 6 1; ...
%K A052174 nonn,tabl,easy,nice
%O A052174 0,4
%A A052174 njas, Jan 26 2000
%I A008276
%S A008276 1,1,1,1,3,2,1,6,11,6,1,10,35,50,24,1,15,85,225,274,120,1,
%T A008276 21,175,735,1624,1764,720,1,28,322,1960,6769,13132,13068,
%U A008276 5040,1,36,546,4536,22449,67284,118124,109584,40320,1,45
%V A008276 1,1,-1,1,-3,2,1,-6,11,-6,1,-10,35,-50,24,1,-15,85,-225,274,-120,1,
%W A008276 -21,175,-735,1624,-1764,720,1,-28,322,-1960,6769,-13132,13068,
%X A008276 -5040,1,-36,546,-4536,22449,-67284,118124,-109584,40320,1,-45
%N A008276 Triangle of Stirling numbers of 1st kind, s(n,n-k+1), n >= 1, 1<=k<=n. Also triangle T(n,k) giving coefficients in expansion of n!*C(x,n)/x in powers of x.
%D A008276 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%D A008276 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
%F A008276 n!*binomial(x,n)= Sum T(n,k)*x^(n-k), k=1..n-1.
%e A008276 3!*C(x,3) = x^3-3*x^2+2*x.
%e A008276 1; 1,-1; 1,-3,2; 1,-6,11,-6; 1,-10,35,-50,24; ...
%o A008276 (PARI) T(n,k)=polcoeff(n!*binomial(x,n),n-k)
%Y A008276 See A008275 and A048994, which are the main entries for this triangle of numbers. Cf. A054654, A054655.
%K A008276 sign,done,tabl,nice
%O A008276 1,5
%A A008276 njas
%I A016556
%S A016556 3,2,1,6,25,1,10,1,4,2,2,3,1,1,2,2,1,40,1,2,3,1,1,2,1,1,
%T A016556 1,4,1,2,1,1,1,1,2,2,1,1,1,5,1,1,4,1,73,1,6,1,6,2,1,11,
%U A016556 1,13,74,1,1,14,1,5,67,1,52,1,2,1,2,3,2,3,1,2,1,1,1,11
%N A016556 Continued fraction for ln(57/2).
%K A016556 nonn,cofr
%O A016556 1,1
%A A016556 njas
%I A001355 M2239 N0889
%S A001355 3,2,1,7,4,1,1,8,5,2,9,8,2,1,6,8,5,2,3,8,5,4,8,5,9,9,7,0,9,4,3,5,2,2,3,
%T A001355 3,8,5,4,3,6,6,2,0,6,2,4,8,3,7,3,4,8,7,3,1,2,3,7,5,9,2,5,6,0,6,2,2,8,4,
%U A001355 8,9,4,7,1,7,9,5,7,7,1,2,6,4,9,7,3,0,9,9,9,3,3,6,7,9,5,9,1,9
%N A001355 Mix digits of pi and e.
%t A001355 Flatten[Transpose[{RealDigits[Pi,10,50][[1]],RealDigits[E,10,50][[1]]}]]
%Y A001355 Cf. A058382.
%K A001355 nonn,base,dumb,easy
%O A001355 1,1
%A A001355 njas
%I A059380
%S A059380 1,1,1,1,3,2,1,7,8,2,1,15,26,12,4,1,31,80,56,24,2,1,63,242,240,
%T A059380 124,24,6,1,127,728,992,624,182,48,4,1,255,2186,4032,3124,1200,
%U A059380 342,48,6,1,511,6560,16256,15624,7502,2400,448,72,4,1,1023,19682
%N A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).
%D A059380 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%e A059380 Array begins:
%e A059380 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, ...
%e A059380 1, 3, 8, 12, 24, 24, 48, 48, 72, 72, ...
%e A059380 1, 7, 26, 56, 124, 182, 342, 448, 702, ...
%e A059380 1, 15, 80, 240, 624, 1200, 2400, 3840, ...
%p A059380 J:=proc(n,k) local i,p,t1,t2; t1:=n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1:=t1*(1-p^(-k)); fi; od; t1; end;
%Y A059380 See A059379 and A059390 (triangle of values of J_k(n)); A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.
%K A059380 nonn,tabl
%O A059380 1,5
%A A059380 njas, Jan 28 2001
%I A056151
%S A056151 1,1,1,1,3,2,1,7,10,6,1,15,38,42,24,1,31,130,222,216,120,1,63,422,1050,
%T A056151 1464,1320,720,1,127,1330,4686,8856,10920,9360,5040,1,255,4118,20202,
%U A056151 50424,80520,91440,75600,40320,1,511,12610,85182,276696,558120,795600
%N A056151 Distribution of maximum inversion table entry.
%D A056151 "An Introduction to the Analysis of Algoritms", R. Sedgewick and Ph. Flajolet, Addison-Wesley,1996, ISBN 0-201-40009-X, table 6.10 (page 356)
%F A056151 Table[ -((-1 + k)^(1 - k + n)*(-1 + k)!) + k^(-k + n)*k!, {n, 1, 9}, {k, 1, n} ]
%e A056151 {1}, {1, 1}, {1, 3, 2}, {1, 7, 10, 6},
%Y A056151 Columns and diagonals give A000225, A056182, A000142, A056197.
%K A056151 nonn,tabl,easy
%O A056151 0,5
%A A056151 Wouter Meeussen (wouter.meeussen@vandemoortele.com), Aug 05 2000
%E A056151 More terms from Larry Reeves (larryr@acm.org), Oct 03 2000
%I A028246
%S A028246 1,1,1,1,3,2,1,7,12,6,1,15,50,60,24,1,31,180,390,360,120,1,63,602,2100,
%T A028246 3360,2520,720,1,127,1932,10206,25200,31920,20160,5040,1,255,6050,
%U A028246 46620,166824,317520,332640,181440,40320,1,511,18660,204630,1020600
%N A028246 Triangular array of numbers a(n,k) = Sum_{i=0..k} (-1)^(k-i)*C(k,i)*i^n; n >= 1, 1<=k<=n.
%C A028246 Let M = n X n matrix with (i,j)-th entry a(n+1-j, n+1-i), e.g. if n = 3, M = [1 1 1; 3 1 0; 2 0 0]. Given a sequence s = [s(0)..s(n-1)], let b = [b(0)..b(n-1)] be its inverse binomial transform, and let c = [c(0)..c(n-1)] = M^(-1)*transpose(b). Then s(k) = Sum_{i=0..n-1) b(i)*binomial(k,i) = Sum_{i=0..n-1) c(i)*k^i, k=0..n-1. - Gary W. Adamson, Nov 11, 2001.
%H A028246 N. J. A. Sloane, Transforms
%F A028246 a(n,k) = S2(n,k)*(k-1)! where S2(n,k) is a Stirling number of the second kind (cf. A008277).
%e A028246 1; 1,1; 1,3,2; 1,7,12,6; 1,15,50,60,24; ...
%e A028246 Row 5 of triangle is {1,15,50,60,24}, which is {1,15,25,10,1} times {0!,1!,2!,3!,4!}.
%Y A028246 Dropping the column of 1's gives A053440. Essentially same as A008277.
%K A028246 nonn,easy,nice,tabl
%O A028246 1,5
%A A028246 njas, Doug McKenzie (mckfam4@aol.com)
%E A028246 More terms from James A. Sellers (sellersj@math.psu.edu), Jan 14 2000
%I A016648
%S A016648 3,2,1,8,8,7,5,8,2,4,8,6,8,2,0,0,7,4,9,2,0,1,5,1,8,6,6,6,4,5,2,3,7,
%T A016648 5,2,7,9,0,5,1,2,0,2,7,0,8,5,3,7,0,3,5,4,4,3,8,2,5,2,9,5,7,8,2,9,4,
%U A016648 8,3,5,7,9,7,5,4,1,5,3,1,5,5,2,9,2,6,0,2,6,7,7,5,6,1,8,6,3,5,9,2,2
%N A016648 Decimal expansion of ln(25).
%D A016648 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
%K A016648 nonn,cons
%O A016648 1,1
%A A016648 njas
%I A002350 M2240 N0890
%S A002350 1,1,3,2,1,9,5,8,3,1,19,10,7,649,15,4,1,33,17,170,9,55,197,24,5,1,51,
%T A002350 26,127,9801,11,1520,17,23,35,6,1,73,37,25,19,2049,13,3482,199,161
%N A002350 Solution to Pellian: x such that x^2 - n y^2 = 1.
%D A002350 L. Beeckmans, Squares expressible as sum of consecutive squares, Amer. Math. Monthly, 101 (1994), 437-442.
%D A002350 A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
%D A002350 C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
%D A002350 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
%Y A002350 Cf. A002349, A006702-A006705.
%K A002350 nonn,nice,easy
%O A002350 1,3
%A A002350 njas
%I A057731
%S A057731 1,1,1,1,3,2,1,9,8,6,1,25,20,30,24,20,1,75,80,180,144,240,
%T A057731 1,231,350,840,504,1470,720,0,0,504,0,420,1,763,1232,5460,1344,10640,5760,
%U A057731 5040,0,4032,0,3360,0,0,2688,1,2619,5768,30996,3024,83160,25920,45360,40320,27216,0,30240,0,25920,24192,0,0,0,0,18144
%N A057731 Table T(n,k) giving number of elements of order k in symmetric group S_n, n >= 1, 1<=k<=g(n), where g(n) = A000793(n) is Landau's function..
%e A057731 1; 1,1; 1,3,2; 1,9,8,6; 1,25,20,30,24,20; ...
%o A057731 (Magma) {* Order(g) : g in Sym(6) *};
%Y A057731 Cf. A054522 (for cyclic group), A057740 (alternating group), A057741 (dihedral group).
%K A057731 nonn,tabf,easy,nice
%O A057731 1,5
%A A057731 Roger CUCULIERE (cuculier@imaginet.fr), Oct 29 2000
%E A057731 More terms from njas, Nov 01 2000
%I A059418
%S A059418 1,1,1,3,2,1,12,7,4,1,60,33,19,7,1,360,192,109,47,11,1,2520,1320,737,
%T A059418 344,102,16,1,20160,10440,5742,2801,956,198,22,1,181440,93240,50634,
%U A059418 25349,9493,2342,352,29,1,1814400,927360,498312,253426,101293,28229
%N A059418 Triangle T(n,k) arising from enumeration of permutations with ordered orbits, read by rows (1<=k<=n).
%D A059418 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 258, #10, F(n,k).
%F A059418 T(n,k) = (n-2)*T(n-1,k) + T(n-1,k-1), T(n,1)=n!/2, T(n,n)=1.
%e A059418 1; 1,1; 3,2,1; 12,7,4,1; 60,33,19,7,1; ...
%Y A059418 Diagonals give A001710, A006595.
%K A059418 nonn,easy,tabl
%O A059418 1,4
%A A059418 njas, Jan 30 2001
%E A059418 More terms from Larry Reeves (larryr@acm.org), Jan 31 2001
%I A048647
%S A048647 0,3,2,1,12,15,14,13,8,11,10,9,4,7,6,5,48,51,50,49,60,63,62,61,56,59,58,
%T A048647 57,52,55,54,53,32,35,34,33,44,47,46,45,40,43,42,41,36,39,38,37,16,19,
%U A048647 18,17,28,31,30,29,24,27,26,25,20,23,22,21,192,195,194,193,204,207,206
%N A048647 Write n in base 4, then replace each digit by its base 4 negative.
%C A048647 The graph of a(n) on [ 1..4^k ] resembles a plane fractal of fractal dimension 1.
%C A048647 Self-inverse considered as a permutation of the positive integers.
%H A048647 J. W. Layman, View fractal-like graph
%H A048647 Index entries for sequences that are permutations of the natural numbers
%e A048647 a(15)=5, since 15 = 33(base 4) -> 11(base 4) = 5.
%Y A048647 Cf. A065256.
%K A048647 nonn,easy,nice
%O A048647 0,2
%A A048647 John W. Layman (layman@math.vt.edu (7/5/99))
%I A059438
%S A059438 1,1,1,3,2,1,13,7,3,1,71,32,12,4,1,461,177,58,18,5,1,3447,1142,327,92,
%T A059438 25,6,1,29093,8411,2109,531,135,33,7,1,273343,69692,15366,3440,800,188,
%U A059438 42,8,1,2829325,642581,125316,24892,5226,1146,252,52,9,1
%N A059438 Triangle T(n,k) (1<=k<=n) read by rows: T(n,k) = no. of permutations of [1..n] with k components.
%D A059438 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).
%F A059438 Let f(x) = Sum_{n >= 0} n!*x^n, g(x) = 1-1/f(x). Then g(x) is g.f. for first diagonal A003319 and Sum_{n >= k} T(n,k)*x^n = g(x)^k.
%e A059438 1; 1,1; 3,2,1; 13,7,3,1; ...
%Y A059438 Diagonals give A003319, A059439, A059440, A055998.
%K A059438 nonn,tabl,easy,nice
%O A059438 1,4
%A A059438 njas, Feb 01 2001
%E A059438 More terms from Vladeta Jovovic (vladeta@Eunet.yu), Mar 04 2001
%I A010269
%S A010269 3,2,2,1,1,1,1,1,7,1,2,3,1,1,1,1,4,1,1,8,2,7,3,1,1525,1,
%T A010269 4,4,2,2,7,1,2,6,1,1,2,3,27,1,83,1,1,1,1,5,1,11,1,15,7,
%U A010269 16,1,2,4,4,1,1,4,1,2,7,1,1,5,1,1,3,1,1,4,1,1,23,1,6,7
%N A010269 Continued fraction for cube root of 40.
%K A010269 nonn,cofr
%O A010269 0,1
%A A010269 njas
%I A054546
%S A054546 1,3,2,2,1,1,2,2,1,1,2,2,1,1,2,1,1,1,1,2,2,1,1,1,1,2,1,1,2,2,1,1,2,1,1,
%T A054546 1,1,2,1,1,1,1,2,2,1,1,1,1,2,1,1,2,2,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,1,1,
%U A054546 1,1,2,1,1,2,2,1,1,2,2,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2
%N A054546 Sum of first n terms equals n-th nonprime number.
%C A054546 First differences of sequence 0, 1, 4, 6, ... (0, 1, composite numbers, see A002808).
%t A054546 t=Flatten[Position[Table[PrimeQ[w],{w,2,256}],False]]+1 Delete[t-RotateRight[t], 1]
%Y A054546 Cf. A018252, A066110, A001223.
%K A054546 nonn
%O A054546 1,2
%A A054546 G. L. Honaker, Jr. (curios@bvub.com), Apr 09 2000
%E A054546 More terms from James A. Sellers (sellersj@math.psu.edu), Apr 11 2000
%I A065310
%S A065310 3,2,2,1,1,2,2,1,1,2,2,1,1,2,1,1,1,1,2,2,1,1,1,1,2,1,1,2,2,1,1,2,1,1,1,
%T A065310 1,2,1,1,1,1,2,2,1,1,1,1,2,1,1,2,2,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,1,1,1,
%U A065310 1,2,1,1,2,2,1,1,2,2,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,2
%N A065310 Number of occurrences of n-th prime in A065308, i.e. in sequence of p[j-Pi[j]]=A000040[j-A000720(j)].
%C A065310 Seems identical to A054546. Each odd primes arise once or twice!?
%t A065310 t=Table[Prime[w-PrimePi[w]],{w,a,b}] Table[Count[t,Prime[n]],{n,c,d}]
%Y A065310 A000040, A000720, A065308-A065309, A054576.
%K A065310 nonn
%O A065310 1,1
%A A065310 Labos E. (labos@ana1.sote.hu), Oct 29 2001
%I A016558
%S A016558 3,2,2,1,1,5,1,30,5,1,1,1,41,1,3,1,6,8,2,1,4,2,5,3,3,376,
%T A016558 4,2,21,4,1,1,13,13,1,1,1,15,1,1,11,2,2,3,1,5,3,1,1,4,1,
%U A016558 1,1,3,14,1,1,1,1,3,1,4,1,1,1,2,1,5,1,1,1,1,18,1,1,1,1
%N A016558 Continued fraction for ln(61/2).
%K A016558 nonn,cofr
%O A016558 1,1
%A A016558 njas
%I A054081
%S A054081 1,3,2,2,1,3,6,5,5,4,8,7,1,6,5,4,3,8,8,7,6,11,10,2,1,9,8,7,5,4,11,11,
%T A054081 11,10,9,8,14,13,13,2,1,12,11,10,9,16,15,4,14,14,14,13,12,11,10,7,6,16,
%U A054081 3,2,1,15,14,13,12,11,19,18,18,17,17,17,17,16
%N A054081 Array T by antidiagonals: for n >= 0 and k >= 1, let p(k)=least number in N not already an a(i), q(k)=p(k)+k+n-1, a(p(k))=q(k), a(q(k))=p(k); then for h>=1, T(n,h)=a(h).
%C A054081 Each row is a self-inverse permutation of N; are the numbers in every column distinct?
%H A054081 Index entries for sequences that are permutations of the natural numbers
%e A054081 Row 0: 1 3 2 6 8 4 11 ... = A019444.
%e A054081 Row 1: 2 1 5 7 3 10 4 ... = A026243.
%e A054081 Row 2: 3 5 1 8 2 11 13 ...
%K A054081 nonn,tabl,eigen
%O A054081 1,2
%A A054081 Clark Kimberling, ck6@cedar.evansville.edu
%I A050604
%S A050604 1,3,2,2,1,4,3,3,1,3,2,2,1,5,4,4,1,3,2,2,1,4,3,3,1,3,2,
%T A050604 2,1,6,5,5,1,3,2,2,1,4,3,3,1,3,2,2,1,5,4,4,1,3,2,2,1,4,
%U A050604 3,3,1,3,2,2,1,7,6,6,1,3,2,2,1,4,3,3,1,3,2,2,1,5,4,4,1
%N A050604 Column 3 of A050600: a(n) = add1c(n,3).
%K A050604 nonn
%O A050604 0,2
%A A050604 Antti.Karttunen@iki.fi (karttu@megabaud.fi) 22-JUN-1999
%I A058758
%S A058758 1,0,1,1,1,0,3,2,2,2,2,2,5
%N A058758 McKay-Thompson series of class 84A for Monster.
%D A058758 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058758 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058758 nonn
%O A058758 -1,7
%A A058758 njas, Nov 27, 2000
%I A057934
%S A057934 1,1,3,2,2,2,2,2,5,3,5,3,3,4,7,5,4,3,2,4,8,4,5,3,5,3,7,4,3,7,2,4,9,4,5,
%T A057934 6,4,3,10,4,3,7,4,4,12,4,4,9,4,7,8,4,2,6,10,5,6,5,4,6,3,3,12,3,6,8,2,4,
%U A057934 10,11,3,5,4,7,11,6,12,7,4,9,11,3,7,8,8,3,8,4,4,11,6,4,8,4,6,8,4,5,13
%N A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).
%C A057934 2^(a(2n)-1)-1 predicts the number of pair_solutions of even length L for AB = A^2 + B^2. For instance with length 18 we have 10^18 + 1 = 101.9901.999999000001 or 3 divisors F which when put into the Mersenne formula 2^(F-1)-1 yields 3 pairs (see ref. 'Puzzle 104' for details).
%H A057934 C. Rivera, Puzzle 104
%H A057934 S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
%H A057934 S. S. Wagstaff, Jr., The Cunningham Project
%Y A057934 Cf. A003021, A001271, A046053, A057935-A057941, A054992, A001562.
%K A057934 nonn
%O A057934 1,3
%A A057934 Patrick De Geest (pdg@worldofnumbers.com), Oct 2000.
%I A037199
%S A037199 3,2,2,2,2,3,4,3,3,2,3,2
%N A037199 Number of consonants in n (in German).
%K A037199 nonn,word,easy,more
%O A037199 0,1
%A A037199 njas
%I A064123
%S A064123 3,2,2,2,4,2,2,2,4,2,2,2,2,4,4,4,4,2,8,4,4,8,4,2,16,4,16,2,8,4,4,4,4,4,
%T A064123 16,4,8,8,4,4,4,8,4,4,8,4,2,2,2,4,8,4,8,8,16,2,2,4,4
%N A064123 Number of divisors of 5^n - 1 that are relatively prime to 5^m - 1 for all 0 < m < n.
%H A064123 Sam Wagstaff, Cunningham Project, Factorizations of 5^n-1, n odd, n<376
%t A064123 a = {1}; Do[ d = Divisors[ 5^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 5^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 58} ]
%Y A064123 Cf. A063982.
%K A064123 nonn
%O A064123 1,1
%A A064123 Robert G. Wilson v (rgwv@kspaint.com), Sep 10 2001
%I A024703
%S A024703 0,1,1,1,3,2,2,2,4,2,3,2,3,3,2,2,2,4,3,4,2,4,5,3,3,3,4,4,7,3,3,3,3,4,2,4,3,
%T A024703 2,3,4,6,5,4,5,3,5,2,3,4,5,5,5,7,3,4,6,2,4,2,3,4,4,3,2,3,5,2,4,5,4,4,3,5,6,
%U A024703 3,3,5,4,4,3,6,6,4,3,8,3,4,5,3,5,6,4,4,4,3,5,3,4,3,3,2,4,4,7,3,6,5,5,5,3,4
%N A024703 Prime divisors, including repetitions, of n-th term of A024702.
%K A024703 nonn
%O A024703 1,5
%A A024703 Clark Kimberling (ck6@cedar.evansville.edu)
%I A064126
%S A064126 3,2,2,2,4,4,4,4,2,2,4,2,8,2,4,4,4,4,2,4,8,8,2,2,8,4,4,8,32,8,8,32,4,8,
%T A064126 8,2,8,2,2,4,16,8,16,8,4,16,4,2,4,8,16,4,16,4,16,4,8,4,4,8,128
%N A064126 Number of divisors of 10^n - 1 that are relatively prime to 10^m - 1 for all 0 < m < n.
%H A064126 Sam Wagstaff, Cunningham Project, Factorizations of 10^n-1, n odd, n<300
%t A064126 a = {1}; Do[ d = Divisors[ 10^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 10^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 61} ]
%Y A064126 Cf. A063982.
%K A064126 nonn
%O A064126 1,1
%A A064126 Robert G. Wilson v (rgwv@kspaint.com), Sep 10 2001
%I A065437
%S A065437 3,2,2,2,10,3,2,18,3,4,2,6,5,6,46,52,4,6,5,7,2,78,5,8,8,10,102,2,5,8,2,
%T A065437 10,136,138,148,3,7,162,166,3,178,10,6,12,6,11,8,222,8,12,3,14,12,8,2,
%U A065437 262,268,7,11,14,282,292,7,310,2,316,15,9,346,8,10,358,366,4,13,10,388
%N A065437 Smallest base relative to which the n-th prime is palindromic.
%C A065437 Subset of A016026 for primes only.
%H A065437 Eric W. Weisstein, Palindromic Primes
%e A065437 71 is the 20-th prime, and can be written as 131 in base 7, hence a(20)=7
%t A065437 PrimeMinBase[ n_ ] := NestWhile[ # + 1 &, 2, IntegerDigits[ Prime[ n ], # ] != Reverse[ IntegerDigits[ Prime[ n ], # ] ] & ]
%Y A065437 Cf. A016026.
%K A065437 base,easy,nonn
%O A065437 1,1
%A A065437 Peter Bertok (peter@bertok.com), Nov 23 2001
%I A006379 M2241
%S A006379 0,0,0,0,1,1,1,1,3,2,2,3,0,4,3,1,2,2,0,1,2
%N A006379 Non-cyclic simple groups with n conjugacy classes.
%D A006379 C. A. Landauer, Simple Groups with 9, 10 and 11 Conjugate Classes. Ph.D. Dissertation, California Inst. Tech., Pasadena, 1973.
%H A006379 Index entries for sequences related to groups
%K A006379 nonn,nice
%O A006379 1,9
%A A006379 njas
%I A052901
%S A052901 3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,
%T A052901 2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,
%U A052901 2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2
%N A052901 A simple periodic sequence: a(3n)=3, a(3n+1)=a(3n+2)=2.
%H A052901 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 878
%F A052901 G.f.: -(2*x^2+2*x+3)/(-1+x^3)
%F A052901 Sum(1/3*(2*_alpha^2+3*_alpha+2)*_alpha^(-1-n),_alpha=RootOf(-1+_Z^3))
%p A052901 spec:= [S,{S=Union(Sequence(Z),Sequence(Z),Sequence(Prod(Z,Z,Z)))},unlabelled]: seq(combstruct[count](spec,size=n),n=0..20);
%K A052901 easy,nonn
%O A052901 0,1
%A A052901 encyclopedia@pommard.inria.fr, Jan 25 2000
%E A052901 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 06 2000
%I A049234
%S A049234 3,2,2,3,2,4,2,4,3,2,4,4,2,6,3,4,2,6,4,2,6,5,4,4,6,2,8,2,4,4,4,4,2,4,2,
%T A049234 6,4,8,3,6,2,4,2,8,2,4,4,9,2,8,4,2,6,4,4,4,2,8,6,2,8,4,4,2,12,4,6,4,2,
%U A049234 8,4,3,6,8,4,8,4,8,4,4,2,6,2,8,9,4,4,8,2,8,4,4,4,4,4,4,4,2,12,4,6,4,4
%N A049234 Number of divisors of p_k + 2, where p_k is k-th prime.
%F A049234 DivisorSigma[ 0,Prime[ n ]+2 ]
%e A049234 10th term is a(10)=d[ p(10)+2 ]=d[ 29+2 ]=d[ 29 ]=2
%K A049234 nonn
%O A049234 1,1
%A A049234 Labos E. (labos@ana1.sote.hu)
%I A032536
%S A032536 1,1,3,2,2,3,3,2,11,10,9,9,8,8,8,7,7,11,10,10,10,9,9,9,8,8,37,35,34,
%T A032536 33,32,31,30,30,29,30,29,29,28,27,27,26,26,25,26,26,25,25,24,24,23,
%U A032536 23,23,37,36,35,35,34,34,33,33,32,33,32,32,31,31,31,30,30,29,30,30
%N A032536 Integer part of decimal 'base 3 looking' numbers divided by their actual base 3 values.
%Y A032536 Cf. A032537, A032538. See also A032532 for explanation.
%K A032536 nonn,easy
%O A032536 0,3
%A A032536 Patrick De Geest (pdg@worldofnumbers.com), april 1998.
%I A046822
%S A046822 1,3,2,2,3,4,1,3,3,5,3,4,5,3,2,4,3,5,4,3,4,5,3,5,5,7,2,3,4,4,3,
%T A046822 5,3,5,4,4,5,6,2,4,4,6,4,5,6,5,4,6,5,7,6,2,3,4,2,4,4,6,3,4,5,5,
%U A046822 4,6,3,5,4,4,5,6,3,5,5,7,5,6,7,4,3,5,4,6,5,4,5,6,4,6,6,8,4,5,6
%N A046822 Number of 1's in binary expansion of 5n+2.
%K A046822 nonn
%O A046822 0,2
%A A046822 njas
%I A029246
%S A029246 1,0,0,1,1,1,1,1,3,2,2,3,4,4,4,5,7,6,7,8,10,10,10,12,15,
%T A029246 14,15,17,20,20,21,23,27,27,28,31,35,35,37,40,45,45,47,
%U A029246 51,56,57,59,63,70,70,73,78,84,86,89,94,102,103,107,113
%N A029246 Expansion of 1/((1-x^3)(1-x^4)(1-x^5)(1-x^8)).
%K A029246 nonn
%O A029246 0,9
%A A029246 njas
%I A059942
%S A059942 1,1,3,2,2,3,7,4,5,4,4,5,4,7,15,8,9,8,9,10,9,8,8,9,10,9,8,9,8,15,31,16,
%T A059942 17,16,19,16,17,16,17,18,21,16,17,18,17,16,16,17,18,17,16,21,18,17,16,
%U A059942 17,16,19,16,17,16,31,63,32,33,32,35,32,33,32,35,36,33,32,35,32,33,32
%N A059942 A059941 translated from binary to decimal.
%F A059942 a(n) =A059943(n)/2
%e A059942 a(35)=19 since A059941(35)=10011.
%K A059942 base,nonn
%O A059942 0,3
%A A059942 Henry Bottomley (se16@btinternet.com), Feb 14 2001
%I A032450
%S A032450 1,3,2,2,3,7,6,12,4,2,3,12,4,7,6,4,7,6,12,15,8,12,28,6,12,4,7,12,4,7,6,
%T A032450 28,12,6,12,4,7,8,15,8,15,30,72,24,60,16,31,6,12,4,7,24,60,16,31,30,72,
%U A032450 8,15,12,28
%N A032450 String of periods of finite sequences related to Poulet's Conjecture.
%D A032450 Alaoglu and Erdos: A conjecture... Bull. Amer Math. Soc. 50 (1944) 881-882
%F A032450 a(1)=n,a(2k)=sigma(a(2k-1)),a(2k+1)=phi(a(2k)), n=1..inf (definition)
%K A032450 nonn
%O A032450 0,2
%A A032450 Ursula Gagelmann (gagelmann@altavista.net)
%I A046460
%S A046460 1,3,2,2,3,8,2,5,4,3,6,6,3,3,3,4,5,6,6,10,8,6,4,6,5,9,8,7,4,7,3,6,6,2,
%T A046460 8,9,4,4,6,9,5,7,4,7,10,7,5,8,6,10,3,9,8,14,5,5,6,4,4,8,3,8,5,10
%N A046460 Number of prime factors of concatenation of numbers from 1 up to n, with multiplicity.
%D A046460 M. Fleuren, Smarandache Factors and Reverse Factors, Smarandache Notions Journal, 10 (No. 1-2-3, Spring 1999), 5-38.
%H A046460 P. De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
%H A046460 M. Fleuren, Smarandache factors
%H A046460 M. L. Perez et al., eds., Smarandache Notions Journal
%H A046460 C. Rivera, Primes by Listing
%Y A046460 Cf. A007908, A046461-A046468, A050677.
%K A046460 nonn,hard,base
%O A046460 0,2
%A A046460 Patrick De Geest (pdg@worldofnumbers.com), Aug 1998.
%I A058743
%S A058743 1,0,1,1,1,1,3,2,2,4,4,4,7,6,8,10,10,11,16,15,18,22,24,26,34
%N A058743 McKay-Thompson series of class 69A for Monster.
%D A058743 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058743 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058743 nonn
%O A058743 -1,7
%A A058743 njas, Nov 27, 2000
%I A024938
%S A024938 3,2,2,5,1,5,3,5,5,7,5,10,6,10,12,10,15,12,16,17,17,19,22,17,27,21,30,30,31
%N A024938 a(n) = total number of parts in all partitions of n into distinct primes.
%K A024938 nonn
%O A024938 7,1
%A A024938 Clark Kimberling (ck6@cedar.evansville.edu)
%I A058608
%S A058608 1,1,1,1,3,2,2,5,6,7,7,9,12
%V A058608 1,-1,1,-1,3,-2,2,-5,6,-7,7,-9,12
%N A058608 McKay-Thompson series of class 28C for Monster.
%D A058608 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058608 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058608 sign,done
%O A058608 -1,5
%A A058608 njas, Nov 27, 2000
%I A021035
%S A021035 0,3,2,2,5,8,0,6,4,5,1,6,1,2,9,0,3,2,2,5,8,0,6,4,5,1,6,1,2,9,0,3,2,
%T A021035 2,5,8,0,6,4,5,1,6,1,2,9,0,3,2,2,5,8,0,6,4,5,1,6,1,2,9,0,3,2,2,5,8,
%U A021035 0,6,4,5,1,6,1,2,9,0,3,2,2,5,8,0,6,4,5,1,6,1,2,9,0,3,2,2,5,8,0,6,4
%N A021035 Decimal expansion of 1/31.
%K A021035 nonn,cons
%O A021035 0,2
%A A021035 njas
%I A007567 M2242
%S A007567 3,2,2,5,11,59,659,38939,25661459,999231590939,
%T A007567 25641740502411581459,25622037156669717708454796390939,
%U A007567 656993627914472375437286314449293585586011019581459
%V A007567 -3,-2,2,5,11,59,659,38939,25661459,999231590939,
%W A007567 25641740502411581459,25622037156669717708454796390939,
%X A007567 656993627914472375437286314449293585586011019581459
%N A007567 Knopfmacher expansion of 1/2: a(n+1) = a(n-1)(a(n)+1)-1.
%D A007567 A. Knopfmacher, ``Rational numbers with predictable Engel product expansions,'' in G. E. Bergum et al., eds., Applications of Fibonacci Numbers. Vol. 5, pp. 421-427.
%H A007567 Index entries for sequences related to Engel expansions
%K A007567 sign,done,easy,nice
%O A007567 0,1
%A A007567 njas,jos
%E A007567 More terms from Christian G. Bower (bowerc@usa.net), Oct 1999.
%I A065474
%S A065474 0,3,2,2,6,3,4,0,9,8,9,3,9,2,4,4,6,7,0,5,7,9,5,3,1,6,9,2,5,4,8,2,3,
%T A065474 7,0,6,6,5,7,0,9,5,0,5,7,9,6,6,5,8,3,2,7,0,9,9,6,1,8,1,1,2,5,2,4,5,
%U A065474 3,2,5,0,0,6,3,4,8,6,2,4,4,6,0,9,8,8,4,5,2,3,4,8,1,5,6,8,5,6,3,7,5
%N A065474 Decimal expansion of product(1 + 2/p^2), p prime >= 2).
%H A065474 G. Niklasch, Some number theoretical constants: 1000-digit values
%e A065474 0.322634098939244670579531692548...
%Y A065474 Cf. A065493.
%K A065474 cons,nonn
%O A065474 1,2
%A A065474 njas, Nov 19 2001
%E A065474 I would also like to get the continued fraction expansion of this number.
%I A021760
%S A021760 0,0,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,
%T A021760 3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,
%U A021760 7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1,3,2,2,7,5,1
%N A021760 Decimal expansion of 1/756.
%K A021760 nonn,cons
%O A021760 0,4
%A A021760 njas
%I A020835
%S A020835 1,1,3,2,2,7,7,0,3,4,1,4,4,5,9,5,7,5,0,6,1,0,7,7,4,7,0,8,5,8,4,5,1,
%T A020835 9,7,3,8,9,9,6,7,4,8,6,5,1,4,0,1,4,6,3,8,2,3,8,7,3,1,0,2,9,4,5,1,9,
%U A020835 9,6,3,3,3,2,5,4,1,4,8,0,0,3,6,4,1,5,0,0,5,5,8,3,8,3,8,6,5,0,9,1,6
%N A020835 Decimal expansion of 1/sqrt(78).
%K A020835 nonn,cons
%O A020835 0,3
%A A020835 njas
%I A055674
%S A055674 0,0,3,2,2,7,16,5,5,12,21,17,9,4,3,44,22,6,8,9,43,38,55,16,10,21,38,31,
%T A055674 40,25,78,12,6,40,48,30,74,58,65,66,24,14,103,34,31,123,71,41,131,27,
%U A055674 114,108,32,84,188,49,74,13,96,130,85,165,94
%N A055674 a(n) = least nonnegative integer h such that (n,h) is not collinear with any 2 points in the set S(n-1):= {(m,a(m)), (m,b(m)): m = 0,1,...,n-1} for n >= 1.
%e A055674 The first five (a(n),b(n)) are (0,1), (0,1), (3,4), (2,7), (2,14).
%Y A055674 Cf. A055675, A055676.
%K A055674 nonn
%O A055674 0,3
%A A055674 Clark Kimberling, ck6@cedar.evansville.edu, Jun 08 2000
%I A011319
%S A011319 1,1,3,2,2,9,3,6,2,5,3,4,0,5,3,6,5,8,7,9,9,5,2,5,8,5,5,9,8,7,7,5,8,
%T A011319 7,8,5,0,1,7,6,6,9,3,8,4,9,7,7,4,9,2,6,9,6,0,1,4,8,8,4,3,4,1,6,9,7,
%U A011319 4,5,0,1,7,7,8,6,8,7,1,2,1,9,1,9,2,9,9,5,1,9,4,6,9,4,1,8,2,6,8,5,9
%N A011319 Decimal expansion of 20th root of 12.
%K A011319 nonn,cons
%O A011319 1,3
%A A011319 njas
%I A058147
%S A058147 1,1,1,1,3,2,2,9,10,6,1,30,47,52,26,2,175,283,413,365,135,1,3333,2139,
%T A058147 3630,4597,3155,875,5
%N A058147 Triangle: Monoids of order n with k idempotents, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).
%H A058147 Index entries for sequences related to monoids
%e A058147 1; 1,1; 1,3,2; 2,9,10,6; 1,30,47,52,26; ...
%Y A058147 Row sums give A058133. Main diagonal: A002788(n-1). Columns 1-3: A000001, A058148, A058149.
%K A058147 nonn,tabl,more
%O A058147 1,5
%A A058147 Christian G. Bower (bowerc@usa.net), Nov 14 2000
%I A053370
%S A053370 0,1,3,2,2,19,5,27,3,131,17,7,11,943,4,4,447,13,5035,9,37,118,703,
%T A053370 15371,79,1595,87,11,28,98,10847,6,6,57731,604,63,1637147,13,478763,
%U A053370 20,43331,34,3583111,7,7,21639,36,66436843,8011739,872,15,5699,77
%N A053370 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of x for n == 1 mod 4.
%C A053370 Entries are indexed by values of n from A039955.
%D A053370 R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
%Y A053370 Cf. A053370-A053375.
%K A053370 nonn,easy,nice
%O A053370 0,3
%A A053370 njas, Jan 06 2000
%I A016458
%S A016458 3,2,2,33,160,1,2,51,1,3,22,1,7,1,2,4,7,7,1,1,1,5,4,2,18,
%T A016458 3,12,1,2,1,1,2,1,2,1,1,1,2,1,5,2,7,2,1,2,2,5,4,2,1,1,3,
%U A016458 1,2,2,12,1,1,4,2,1,1,2,2,4,2,2,3,3,2,1,3,2,2,3,1,1,1,11
%N A016458 Continued fraction for ln(30).
%K A016458 nonn,cofr
%O A016458 1,1
%A A016458 njas
%I A058513
%S A058513 1,0,3,2,3,0,0,6,6,8,6,0,16,12,9,10,0,36,21,30,18,0,63,48,53
%V A058513 1,0,3,2,-3,0,0,6,6,-8,6,0,16,12,-9,10,0,36,21,-30,18,0,63,48,-53
%N A058513 McKay-Thompson series of class 15b for Monster.
%D A058513 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058513 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058513 sign,done
%O A058513 -1,3
%A A058513 njas, Nov 27, 2000
%I A047160
%S A047160 0,0,1,0,1,0,3,2,3,0,1,0,3,2,3,0,1,0,3,2,9,0,5,6,3,4,9,0,1,0,9,4,3,6,5,
%T A047160 0,9,2,3,0,1,0,3,2,15,0,5,12,3,8,9,0,7,12,3,4,15,0,1,0,9,4,3,6,5,0,15,
%U A047160 2,3,0,1,0,15,4,3,6,5,0,9,2,15,0,5,12,3,14,9,0,7,12,9,4,15,6,7,0,9,2,3
%N A047160 a(2) = a(3) = 0; for n >= 4, a(n) = smallest number m such that n-m and n+m are both primes, or -1 if no such m exists.
%e A047160 16-3=13 and 16+3=19 are primes, so a(16)=3.
%o A047160 (UBASIC) 10 N=2// 20 M=0// 30 if and{prmdiv(N-M)=N-M,prmdiv(N+M)=N+M} then print M;:goto 50// 40 inc M:goto 30// 50 inc N:if N>130 then stop// 60 goto 20
%K A047160 nonn,easy,nice
%O A047160 2,7
%A A047160 Lior Manor (lior_manor@icominfosys.com)
%E A047160 More terms from Patrick De Geest (pdg@worldofnumbers.com), May 1999.
%I A048967
%S A048967 0,0,1,0,3,2,3,0,7,6,7,4,9,6,7,0,15,14,15,12,17,14,15,8,21,18,19,12,21,
%T A048967 14,15,0,31,30,31,28,33,30,31,24,37,34,35,28,37,30,31,16,45,42,43,36,
%U A048967 45,38,39,24,49,42,43,28,45,30,31,0,63,62,63,60,65,62,63,56,69,66,67
%N A048967 Number of even entries in row n of Pascal's triangle (A007318).
%C A048967 In rows 2^k - 1 all entries are odd.
%C A048967 a(n) = 0 (all the entries in the row are odd) iff n = 2^m - 1 for some m >= 0 and then n belongs to sequence A000225. - Avi Peretz (njk@netvision.net.il), Apr 21 2001
%H A048967 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A048967 a(n) = n+1 - A001316(n) = n+1 - 2^A000120(n) = n+1 - Sum_{k=0..n} (C(n,k) mod 2) = Sum_{ k=0..n} ((1 - C(n,k)) mod 2)
%e A048967 Row 4 is 1 4 6 4 1 with 3 even entries so a(4)=3.
%t A048967 Table[n + 1 - Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ], {n, 0, 100} ]
%Y A048967 Cf. A007318, A001316, A000120, A000225.
%K A048967 easy,nonn
%O A048967 0,5
%A A048967 Brian L. Galebach (briang@ProbabilitySports.com)
%I A046818
%S A046818 1,1,3,2,3,1,3,3,3,3,5,2,3,2,4,4,3,3,5,4,5,1,3,3,3,3,5,3,4,3,5,
%T A046818 5,3,3,5,4,5,3,5,5,5,5,7,2,3,2,4,4,3,3,5,4,5,2,4,4,4,4,6,4,5,4,
%U A046818 6,6,3,3,5,4,5,3,5,5,5,5,7,4,5,4,6,6,5,5,7,6,7,1,3,3,3,3,5,3,4
%N A046818 Number of 1's in binary expansion of 3n+1.
%K A046818 nonn
%O A046818 0,3
%A A046818 njas
%I A033093
%S A033093 0,1,1,3,2,3,1,5,5,5,3,6,2,3,3,8,5,9,5,8,5,4,2,9,5,5,7,9,5,8,
%T A033093 2,11,9,8,8,13,6,7,6,11,5,9,3,7,8,5,3,13,7,10,8,9,5,12,7,11,6,
%U A033093 5,3,13,3,4,6,15,12,14,8,11,9,12,6,18,8,9,11,11,9,11,5,14,13
%N A033093 Number of 0's when n is written in base b for 2<=b<=n+1.
%K A033093 nonn,base
%O A033093 1,4
%A A033093 Clark Kimberling, ck6@cedar.evansville.edu
%I A064654
%S A064654 1,3,2,3,2,2,2,4,3,1,2,2,2,3,2,2,3,3,2,5,2,1,2,2,2,2,2,4,3,4,3,1,1,2,2,1,
%T A064654 2,3,2,2,1,2,2,2,2,5,3,1,2,2,2,1,2,4,2,2,3,3,2,2,3,1,2,4,1,2,3,1,2,3,3,
%U A064654 4,2,2,3,5,1,2,2,1,1,3,3,1,2,2,2,4,2,2,5,1,1,2,3,1,1,5,2,4,1,3,3,3,1,1
%N A064654 Length of n-th run of evens or odds in A064413.
%D A064654 J. C. Lagarias, E. M. Rains and N. J. A. Sloane, On the EKG sequence, preprint, 2001.
%H A064654 Index entries for sequences related to EKG sequence
%Y A064654 Cf. A064413, A064655, A065656.
%K A064654 nonn,easy
%O A064654 1,2
%A A064654 njas, Oct 09 2001
%E A064654 More terms from Matthew M. Conroy (doctormatt@earthlink.net), Oct 16 2001
%I A056564
%S A056564 1,1,3,2,3,2,2,3,1,1,3,1,2,2,1,1,2,2,1,3,1,1,3,1,1,1,0,1,2,2,2,2,1,0,2,
%T A056564 1,2,2,1,0,3,1,2,2,0,1,1,2,0,2,1,1,3,2,1,2,0,1,1,1,1,1,1,0,2,0,2,2,1,1,
%U A056564 2,1,1,2,1,1,1,1,1,2,1,0,3,1,1,1,0,1,2,2,1,1,0,0,1,1,2,3,1,1,2,1,1,1,1
%N A056564 Number of times primes can be produced by taking the absolute difference between the n-th triangular number and another triangular number.
%F A056564 a(n) =A056562(n)+A056563(n) =number of primes (with multiplicity) in {m, m+1, 2m-1, 2m+3}
%e A056564 a(2)=3 because 2nd triangular number is 3, and all three of 3-0=3, 3-1=2, and 6-3=3 are prime [n=2 is the only case where there is multiplicity].
%Y A056564 Cf. A000217, A001227, A056562, A0565643
%K A056564 nonn
%O A056564 0,3
%A A056564 Henry Bottomley (se16@btinternet.com), Jun 27 2000
%I A049071
%S A049071 0,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,
%T A049071 3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,
%U A049071 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3
%V A049071 0,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,
%W A049071 3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,
%X A049071 -2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3,-2,3
%N A049071 Expansion of x*(3-2*x)/(1-x^2).
%D A049071 B. R. Myers, On spanning trees..., SIAM Rev., 17 (1975), 465-474.
%K A049071 sign,done
%O A049071 0,2
%A A049071 njas
%I A029211
%S A029211 1,0,1,0,1,1,1,1,1,1,3,2,3,2,3,4,4,4,4,4,7,6,8,6,8,9,10,
%T A029211 10,10,10,14,13,16,14,16,18,19,20,20,20,25,24,28,26,29,
%U A029211 31,33,34,35,35,41,40,45,43,47,50,52,54,55,56,63,62,68
%N A029211 Expansion of 1/((1-x^2)(1-x^5)(1-x^10)(1-x^11)).
%K A029211 nonn
%O A029211 0,11
%A A029211 njas
%I A018837
%S A018837 0,3,2,3,2,3,4,5,4,5,6,7,6,7,8,9,8,9,10,11,10,11,12,13,12,13,
%T A018837 14,15,14,15,16,17,16,17,18,19,18,19,20,21,20,21,22,23,22,23,
%U A018837 24,25,24,25,26,27,26,27
%N A018837 Steps for knight to reach (n,0) on infinite chess-board.
%F A018837 2[ (n+2)/4 ] if n even, 2[ (n+1)/4 ]+1 if n odd (n >= 8).
%K A018837 nonn,easy
%O A018837 0,2
%A A018837 njas, Marc Le Brun (mlb@well.com)
%I A039639
%S A039639 3,2,3,2,3,7,2,5,3,7,2,19,5,11,3,13,7,31,17,2,37,5,5,11,3,3,13,13,13,7,
%T A039639 2,2,17,17,37,19,79,41,5,43,11,11,3,97,3,3,53,7,7,7,29,7,7,31,2,2,67,
%U A039639 17,139,17,71,73,19,19,157,79,83,5,43,43,11,11,23,23,47,3,97,199,3
%N A039639 Fixed point of "k -> k/2 or (k-1)/2 until result is prime", starting with prime(n)+1.
%t A039639 see A039634.
%Y A039639 Cf. A039634-A039645.
%K A039639 nonn
%O A039639 0,1
%A A039639 Wouter Meeussen, w.meeussen.vdmcc@vandemoortele.be
%I A023509
%S A023509 3,2,3,2,3,7,3,5,3,5,2,19,7,11,3,3,5,31,17,3,37,5,7,5,7,17,13,3,11,
%T A023509 19,2,11,23,7,5,19,79,41,7,29,5,13,3,97,11,5,53,7,19,23,13,5,11,7,
%U A023509 43,11,5,17,139,47,71,7,11,13,157,53,83,13,29,7,59,5,23,17,19,3,13
%N A023509 Greatest prime divisor of n-th prime + 1.
%K A023509 nonn
%O A023509 1,1
%A A023509 Clark Kimberling (ck6@cedar.evansville.edu)
%I A039641
%S A039641 3,2,3,2,3,7,5,5,3,2,2,19,11,11,3,7,2,31,17,5,37,5,11,23,13,13,13,7,7,
%T A039641 29,2,17,5,5,19,19,79,41,11,11,23,23,3,97,13,13,53,7,29,29,59,2,61,2,
%U A039641 17,17,17,17,139,71,71,37,5,5,157,5,83,43,11,11,89,23,23,47,3,3,13
%N A039641 Fixed point of "k -> k/2 or (k+1)/2 until result is prime", starting with prime(n)+1.
%t A039641 see A039635.
%Y A039641 Cf. A039634-A039645.
%K A039641 nonn
%O A039641 0,1
%A A039641 Wouter Meeussen, w.meeussen.vdmcc@vandemoortele.be
%I A054263
%S A054263 3,2,3,3,2,2,6,2,1,4,7,0,4,4,12,5,6,2,3,2,6,3,6,2,2,4,3,2,5,0,3,2,1,4,
%T A054263 3,1,10,1,4,0
%N A054263 Number of palindromic triangular numbers with n digits.
%H A054263 P. De Geest, Palindromic Triangulars
%e A054263 a(10) = 4 -> 1264114621,1634004361,5289009825,6172882716.
%Y A054263 Cf. A000217, A003098, A008509, A008510.
%K A054263 nonn,base,hard
%O A054263 1,1
%A A054263 Patrick De Geest (pdg@worldofnumbers.com), Apr 2000.
%I A016459
%S A016459 3,2,3,3,2,13,1,5,3,1,32,1,6,2,2,1,24,2,5,1,6,1,16,1,1,
%T A016459 13,14,2,1,5,4,4,1,3,5,4,19,1,16,1,2,5,2,3,1,7,1,1,1,91,
%U A016459 26,1,16,1,13,1,6,29,2,17,1,8,2,28,1,7,6,5,1,1,1,15,1,1
%N A016459 Continued fraction for ln(31).
%K A016459 nonn,cofr
%O A016459 1,1
%A A016459 njas
%I A060585
%S A060585 0,1,1,3,2,3,3,3,2,6,7,7,5,4,5,7,7,6,6,7,7,7,6,7,5,5,4,12,13,13,15,14,
%T A060585 15,15,15,14,10,11,11,9,8,9,11,11,10,14,15,15,15,14,15,13,13,12,12,13,
%U A060585 13,15,14,15,15,15,14,14,15,15,13,12,13,15,15,14,10,11,11,11,10,11,9,9
%N A060585 A ternary to binary switch.
%C A060585 Write n in base 3, then (working from left to right) if the k-th digit of n is equal to the corresponding digit to the left of the k-th digit of a(n) then 1 is the k-th digit of a(n), otherwise the k-th digit of a(n) is 0, then read result as a base 2 number.
%F A060585 a(n) =2a([n/2])+A060584(n) =A060586(A060587(n)).
%e A060585 a(76)=10 since 76 written in base 3 is 2211, and looking for changing digits produces 1010 which read as if in binary is 10.
%Y A060585 k appears A001316(k) times in the sequence.
%K A060585 base,nonn
%O A060585 0,4
%A A060585 Henry Bottomley (se16@btinternet.com), Apr 04 2001
%I A035093
%S A035093 1,1,1,3,2,3,3,4,3,3,1,2,3,13,7,4,5,2,7,17,15,18,3,6,3,16,1,4,7,20,8,3,
%T A035093 9,5,2,7,1,3,10,3,1,29,7,9,45,8,3,6,35,66,2,20,2,4,25,52,14,34,24,6,10,
%U A035093 22,38,16,20,91,69,12,19,20,21,42,1,5,33,77,1,2,12,61,193,74,40,55,19
%N A035093 Smallest k - dependent on n - such that (n!)*k+1 is prime where k is the subscript of the progressions.
%C A035093 This is one possible generalization of "the least prime problem" for nk+1 arithmetical progression when n is replaced by n!, a special difference.
%H A035093 Index entries for sequences related to primes in arithmetic progressions
%e A035093 a(7)=3 because in progression of 5040k+1 the terms 5042 and 10081 are not prime, and so 15121 is the first prime.
%Y A035093 Analogous case is A034693. Special case for k=1 is A002981.
%K A035093 nonn
%O A035093 1,4
%A A035093 Labos E. (labos@ana1.sote.hu)
%I A061266
%S A061266 1,3,2,3,3,4,4,4,4,5,5,5
%N A061266 Number of squares between consecutive cubes.
%e A061266 a(2)= 3 as there are three squares 9,16 and 25 between 8 and 27. a(3)= 2 as there are two squares 36 and 49 between 27 and 64. ( 64 is not to be counted.)
%K A061266 nonn,base
%O A061266 0,2
%A A061266 Amarnath Murthy (amarnath_murthy@yahoo.com), Apr 24 2001
%I A058680
%S A058680 1,3,2,3,3,5,5,9,8,14,14,23,22
%V A058680 1,-3,-2,-3,-3,-5,-5,-9,-8,-14,-14,-23,-22
%N A058680 McKay-Thompson series of class 44a for Monster.
%D A058680 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058680 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058680 sign,done
%O A058680 -1,2
%A A058680 njas, Nov 27, 2000
%I A007888
%S A007888 1,1,3,2,3,4,1,6,45,86,173,100,2641,48311,717766
%V A007888 1,1,3,2,3,4,1,-6,45,-86,173,-100,2641,-48311,717766
%N A007888 Euler characteristic of mapping class group Gamma_n.
%D A007888 Harer, J., The cohomology of the moduli space of curves, Lecture Notes in Math., Vol. 1337, Springer-Verlag, NY, pp. 138-221
%K A007888 sign,done
%O A007888 1,3
%A A007888 Mira Bernstein [ M.Bernstein@pmms.cam.ac.uk ]
%I A028292
%S A028292 3,2,3,4,3,4,3,4,2,2,6,4,7,7,6,7,6,5,6,4,7,6,7,8,7,8,7,8,6,6,9,8,9,
%T A028292 10,9,10,9,10,8,5,8,7,8,9,8,9,8,9,7,5,8,7,8,9,8,9,8,9,7,6,9,8,9,10,
%U A028292 9,10,9,10,8,5,8,7,8,9,8,9,8,9,7,4,7,6,7,8,7,8,7,8,6,5,8,7,8,9,8,9
%N A028292 Numbers of letters in n (in the Norwegian language Nynorsk).
%Y A028292 Cf. A014656.
%K A028292 nonn
%O A028292 1,1
%A A028292 Torleiv.Klove@ii.uib.no
%I A029150
%S A029150 1,0,1,1,1,1,3,2,3,4,4,4,7,6,8,9,10,10,14,13,16,18,19,20,
%T A029150 25,24,28,31,33,34,41,40,45,49,52,54,62,62,68,73,77,80,
%U A029150 90,90,98,104,109,113,125,126,135,143,149,154,168,170,181
%N A029150 Expansion of 1/((1-x^2)(1-x^3)(1-x^6)(1-x^7)).
%K A029150 nonn
%O A029150 0,7
%A A029150 njas
%I A055923
%S A055923 1,0,1,1,1,1,3,2,3,4,4,6,8,8,10,13,13,20,20,24,26,38,35,51,51,65,67,92,
%T A055923 86,121,117,153,155,209,197,270,262,339,341,444,425,565,555,703,711,
%U A055923 903,884,1135,1128,1397,1430,1766,1757,2193,2214,2691,2762,3344
%N A055923 Partitions of n in which each part occurs a prime number (or 0) times.
%F A055923 EULER transform of b where b has g.f. SUM {k>0} c(k)*x^k/(1-x^k) where c is inverse EULER transform of characteristic function of prime numbers.
%Y A055923 Cf. A000041, A007690, A055922.
%K A055923 nonn,part
%O A055923 0,7
%A A055923 Christian G. Bower (bowerc@usa.net), Jun 23 2000
%I A035634
%S A035634 0,0,0,0,1,0,1,1,1,3,2,3,4,4,7,6,9,10,11,16,15,20,23,25,32,34,41,47,
%T A035634 52,63,68,80,90,101,116,129,147,166,184,210,232,262,292,326,363,405,
%U A035634 450,501,554,617,681,756,834,924,1015,1125,1235,1363,1498,1647,1809
%N A035634 Partitions into parts 5k+2 and 5k+3 with at least one part of each type
%K A035634 nonn,part
%O A035634 1,10
%A A035634 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A035366
%S A035366 0,1,1,1,1,3,2,3,4,5,5,8,8,11,12,15,17,22,23,30,34,40,45,56,61,73,83,
%T A035366 98,109,130,144,169,190,219,246,286,317,365,410,467,521,597,663,754,
%U A035366 841,950,1058,1196,1326,1494,1661,1861,2064,2315,2561,2862,3169,3531
%N A035366 Partitions into parts 4k+2 or 4k+3.
%K A035366 nonn,part
%O A035366 1,6
%A A035366 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A003559
%S A003559 1,3,2,3,4,9,5,10,3,15,8,9,4,21,11,21,6,20,14,5,16,11,12,
%T A003559 6,9,39,4,16,10,6,23,24,20,17,26,27,12,44,29,5,15,63,12,
%U A003559 9,16,69,35,14,9,25,30,39,8,81,41,39,21,60,44,45
%N A003559 Least number m such that 3^m = +- 1 mod 3n + 1.
%K A003559 nonn
%O A003559 0,2
%A A003559 njas
%I A064885
%S A064885 3,2,3,5,2,3,8,5,7,2,3,11,8,13,5,12,7,9,2,3,14,11,19,8,21,13,18,5,17,
%T A064885 12,19,7,16,9,11,2,3,17,14,25,11,30,19,27,8,29,21,34,13,31,18,23,5,22,
%U A064885 17,29,12,31,19,26,7,23,16
%N A064885 Eisenstein array Ei(3,2).
%C A064885 In Eisenstein's notation this is the array for m=3 and n=2; see pp.41-2 of the Eisenstein ref. given for A064881. The array for m=n=1 is A049456.
%C A064885 For n >= 1, the number of entries of row is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 5*A007051(n-1).
%C A064885 The binary tree built from the rationals a(n,m)/a(n,m+1), m=0..2^(n-1), for each row n >= 1 gives the sub-tree of the (Eisenstein-)Stern-Brocot tree in the version of, e.g., Calkin and Wilf (for the ref. see A002487, also for the Wilf link) with root 3/2. The composition rule of this tree is i/j -> i/(i+j), (i+j)/j.
%F A064885 a(n,m)= a(n-1,m/2) if m is even, else a(n,m)= a(n-1,(m-1)/2)+a(n-1,(m+1)/2, a(1,0)=3, a(1,1)=2.
%e A064885 {3,2}; {3,5,2}; {3,8,5,7,2}; {3,11,8,13,5,12,7,9,2}; ...
%e A064885 This binary subtree of rationals is built from 3/2; 3/5,5/2; 3/8,8/5,5/7,7/2; ...
%K A064885 nonn,easy,tabf
%O A064885 1,1
%A A064885 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Oct 19 2001
%I A029618
%S A029618 1,3,2,3,5,2,3,8,7,2,3,11,15,9,2,3,14,26,24,11,2,3,17,40,50,35,13,2,3,
%T A029618 20,57,90,85,48,15,2,3,23,77,147,175,133,63,17,2,3,26,100,224,322,308,
%U A029618 196,80,19,2,3,29,126,324,546,630,504,276,99,21,2,3,32,155,450,870
%N A029618 Numbers in (3,2)-Pascal triangle (by row).
%e A029618 1; 3 2; 3 5 2; 3 8 7 2; 3 11 15 9 2; ...
%K A029618 nonn,tabl
%O A029618 0,2
%A A029618 Mohammad K. Azarian, ma3@cedar.evansville.edu
%E A029618 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A023605
%S A023605 0,1,1,1,3,2,3,5,5,4,6,7,7,9,8,9,11,12,12,12,13,13,16,16,16,17,18,
%T A023605 19,20,21,21,22,24,24,24,26,25,27,29,28,29,30,32,33,32,34,35,36,36,
%U A023605 37,38,39,41,43,41,41,44,43,44,46,47,47,50,50,50,50,51,53,55,54,55
%N A023605 Convolution of A023532 and A023534.
%K A023605 nonn
%O A023605 1,5
%A A023605 Clark Kimberling (ck6@cedar.evansville.edu)
%I A050060
%S A050060 1,3,2,3,5,6,8,13,21,22,24,29,37,58,82,119,201,202,204,209,217,238,262,
%T A050060 299,381,582,786,1003,1265,1646,2432,3697,6129,6130,6132,6137,6145,
%U A050060 6166,6190,6227,6309,6510,6714,6931,7193,7574,8360
%N A050060 a(n)=a(n-1)+a(m), where m=2n-3-2^(p+1), and 2^p= 4.
%K A050060 nonn
%O A050060 1,2
%A A050060 Clark Kimberling, ck6@cedar.evansville.edu
%I A051701
%S A051701 3,2,3,5,13,11,19,17,19,31,29,41,43,41,43,47,61,59,71,73,71,83,79,83,
%T A051701 101,103,101,109,107,109,131,127,139,137,151,149,151,167,163,167,181,
%U A051701 179,193,191,199,197,199,227,229,227,229,241,239,257,251,257,271,269
%N A051701 Closest prime to n-th prime p that is different from p (break ties by taking the smaller prime).
%e A051701 Closest primes to 2,3,5,7,11 are 3,2,3,5,13.
%Y A051701 Related sequences: A023186-A023188, A046929-A046931, A051650, A051652, A051697-A051702, A051728-A051730.
%K A051701 nonn,easy,nice
%O A051701 0,1
%A A051701 njas
%E A051701 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A021313
%S A021313 0,0,3,2,3,6,2,4,5,9,5,4,6,9,2,5,5,6,6,3,4,3,0,4,2,0,7,1,1,9,7,4,1,
%T A021313 1,0,0,3,2,3,6,2,4,5,9,5,4,6,9,2,5,5,6,6,3,4,3,0,4,2,0,7,1,1,9,7,4,
%U A021313 1,1,0,0,3,2,3,6,2,4,5,9,5,4,6,9,2,5,5,6,6,3,4,3,0,4,2,0,7,1,1,9,7
%N A021313 Decimal expansion of 1/309.
%K A021313 nonn,cons
%O A021313 0,3
%A A021313 njas
%I A033771
%S A033771 1,1,1,3,2,3,6,5,4,9,9,6,15,12,9,21,19,12,26,24,18,39,31,
%T A033771 24,46,41,28,57,47,30,72,54,38,84,65,48,95,73,53,108,84,
%U A033771 63,134,101,68,147,112,78,173,123,90
%N A033771 Product t2(q^d); d | 12, where t2 = theta2(q)/(2*q^(1/4)).
%K A033771 nonn
%O A033771 0,4
%A A033771 njas
%I A033795
%S A033795 1,1,1,3,2,3,6,5,4,10,10,7,18,14,12,27,24,16,36,34,25,57,
%T A033795 45,36,73,65,44,94,82,56,132,101,77,163,135,96,197,163,
%U A033795 114,252,195,146,312,250,172,362,291,205,452,339,254
%N A033795 Product t2(q^d); d | 36, where t2 = theta2(q)/(2*q^(1/4)).
%K A033795 nonn
%O A033795 0,4
%A A033795 njas
%I A033783
%S A033783 1,1,1,3,2,3,6,5,5,10,10,9,17,15,15,26,23,21,35,30,33,51,
%T A033783 40,45,67,55,56,87,69,75,115,88,93,144,112,120,174,130,
%U A033783 149,209,164,180,260,203,214,310,236,255,370,282,308
%N A033783 Product t2(q^d); d | 24, where t2 = theta2(q)/(2*q^(1/4)).
%K A033783 nonn
%O A033783 0,4
%A A033783 njas
%I A033807
%S A033807 1,1,1,3,2,3,6,5,5,10,10,9,17,15,15,26,24,22,36,33,35,54,
%T A033807 46,50,72,65,66,96,86,90,130,114,116,165,147,150,207,181,
%U A033807 189,254,231,235,316,290,283,385,351,343,465,428,422
%N A033807 Product t2(q^d); d | 48, where t2 = theta2(q)/(2*q^(1/4)).
%K A033807 nonn
%O A033807 0,4
%A A033807 njas
%I A058691
%S A058691 1,1,1,3,2,3,6,5,7,12,12,15,21
%N A058691 McKay-Thompson series of class 48A for Monster.
%D A058691 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058691 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058691 nonn
%O A058691 -1,4
%A A058691 njas, Nov 27, 2000
%I A022472
%S A022472 0,1,3,2,3,6,6,6,12,13,12,23,29,33,50,69,76,108,150,172,226,323,385,
%T A022472 518,698,884,1146,1539
%N A022472 Number of 1's in n-th term of A022470.
%K A022472 nonn
%O A022472 1,3
%A A022472 Clark Kimberling (ck6@cedar.evansville.edu)
%I A014679
%S A014679 1,0,1,3,2,3,6,6,7,10,11,13,16,17,20,24,25,28,33,35,38,
%T A014679 43,46,50,55,58,63,69,72,77,84,88,93,100,105,111,118,123,
%U A014679 130,138,143,150,159,165,172,181,188,196,205,212,221,231
%N A014679 Poincare series for mod 2 cohomology of M_12.
%D A014679 A. Adem, Recent developments in the cohomology of finite groups, Notices Amer. Math. Soc., 44 (1997),806-812.
%D A014679 Alejandro Adem; John Maginnis; James R. Milgram, The geometry and cohomology of the Mathieu group M_12, J. Algebra 139 (1991), no. 1, 90-133.
%p A014679 (1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4));
%K A014679 nonn,easy,nice
%O A014679 0,4
%A A014679 njas
%I A050062
%S A050062 1,3,2,3,6,7,10,12,15,16,19,21,24,30,37,47,59,60,63,65,68,74,81,91,103,
%T A050062 118,134,153,174,198,228,265,312,313,316,318,321,327,334,344,356,371,
%U A050062 387,406,427,451,481,518
%N A050062 a(n)=a(n-1)+a(m), where m=n-1-2^p, and 2^p= 4.
%K A050062 nonn
%O A050062 1,2
%A A050062 Clark Kimberling, ck6@cedar.evansville.edu
%I A058533
%S A058533 1,0,3,2,3,6,10,12,15,22,30,36,44,60,78,96,117,150,190,228,
%T A058533 276,340,420,504,603
%V A058533 1,0,3,-2,3,-6,10,-12,15,-22,30,-36,44,-60,78,-96,117,-150,190,-228,
%W A058533 276,-340,420,-504,603
%N A058533 McKay-Thompson series of class 18C for Monster.
%D A058533 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058533 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058533 sign,done
%O A058533 -1,3
%A A058533 njas, Nov 27, 2000
%I A058644
%S A058644 1,0,3,2,3,6,10,12,15,22,30,36,44,60,78,96,117,150,190,228,276,340,
%T A058644 420,504,603
%N A058644 McKay-Thompson series of class 36A for Monster.
%D A058644 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058644 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058644 nonn
%O A058644 -1,3
%A A058644 njas, Nov 27, 2000
%I A049923
%S A049923 1,3,2,3,6,12,24,39,51,138,276,543,1059,2019,3633,5790,7809,21405,
%T A049923 42810,85611,171195,342291,684177,1366878,2729985,5444355,10824504,
%U A049923 21392328,41760069,79442661,142937349,227824365,307267026,842358414
%N A049923 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=2n-2-2^(p+1), and 2^p= 0, 0<=s<=m, arising in computation of certain integrals.
%D A059366 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 167, a(m,s).
%F A059366 T(m+2,s) = (2*m+3)*(T(m+1,s-1)+T(m+1,s)) - 4*(m+1)^2*T(m,s-1). T(m,s) = Sum_{k=0..s} binomial(s,k)*binomial(2*m-2*k,s)*binomial(2*m-2*k-s,m-k).
%e A059366 1; 1,1; 3,2,3; 15,9,9,15; ...
%Y A059366 Main diagonal gives A001757.
%K A059366 tabl,nonn,easy
%O A059366 0,4
%A A059366 njas, Jan 28 2001
%E A059366 More terms from Larry Reeves (larryr@acm.org), Feb 08 2001
%I A059239
%S A059239 1,1,3,2,3,18,1382,12,32553,87734,1047666,2796588,1418184546,23685516,
%T A059239 366420255876,13785346041608,23127963123651,2729316555918,52630543106106954746,
%U A059239 1850522471899932,4699488932936084196918,18241171727016849632292
%N A059239 Numerators of coefficients of asymptotic expansion related to iterated sine function.
%D A059239 N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland, 3rd. ed., 1970; see p. 158, (8.6.4).
%F A059239 (1-2*n)*(-12)^n*Bernoulli(2*n)/(2*n)!.
%p A059239 (1-2*n)*(-12)^n*bernoulli(2*n)/(2*n)!;
%Y A059239 Cf. A059240.
%K A059239 nonn,easy,frac
%O A059239 0,3
%A A059239 njas, Jan 21 2001
%I A004545
%S A004545 1,3,2,4,0,4,7,4,6,3,1,7,7,1,6,7,4,6,2,2,0,4,2,6,2,7,6,6,1,1,5,4,6,
%T A004545 7,2,5,1,2,5,7,5,1,7,4,3,5,3,3,6,6,0,2,7,2,4,2,2,3,5,6,5,0,2,3,1,6,
%U A004545 6
%N A004545 Expansion of sqrt(2) in base 8.
%K A004545 nonn,base,cons
%O A004545 1,2
%A A004545 njas
%I A057038
%S A057038 1,1,3,2,4,2,4,1,3,5,1,3,5,7,2,4,6,8,2,4,6,8,1,3,5,7,9,1,3,5,7,9,11,2,
%T A057038 4,6,8,10,12,2,4,6,8,10,12,1,3,5,7,9,11,13,1,3,5,7,9,11,13,15,2,4,6,8,
%U A057038 10,12,14,16,2,4,6,8,10,12,14,16,1,3,5,7,9
%N A057038 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; each k is an R(i(k),j(k)), and A057038(n)=i(2n).
%C A057038 j(2n)=A057039(n).
%K A057038 nonn
%O A057038 1,3
%A A057038 Clark Kimberling, ck6@cedar.evansville.edu, Jul 30 2000
%I A013633
%S A013633 3,2,4,2,6,4,4,4,6,2,6,4,4,4,6,2,6,4,4,4,10,6,6,6,6,6,8,
%T A013633 2,8,6,6,6,6,6,10,4,4,4,6,2,6,4,4,4,10,6,6,6,6,6,12,6,6,
%U A013633 6,6,6,8,2,8,6,6,6,6,6,10,4,4,4,6,2,8,6,6,6,6,6,10,4,4
%N A013633 nextprime(n)-prevprime(n).
%p A013633 [ seq(nextprime(i)-prevprime(i),i=3..100) ];
%K A013633 nonn
%O A013633 3,1
%A A013633 njas
%I A016559
%S A016559 3,2,4,2,200,3,2,70,51,4,2,2,7,2,12,2,1,1,1,1,1,1,1,23,
%T A016559 80,2,2,1,7,1,2,47,1,1,1,12,1,9,2,4,1,8,1,1,1,6,1,2,2,2,
%U A016559 4,1,11,3,7,3,1,2,1,7,1,2,1,1,2,5,1,74,1,41,22,2,21,1,1
%N A016559 Continued fraction for ln(63/2).
%K A016559 nonn,cofr
%O A016559 1,1
%A A016559 njas
%I A025509
%S A025509 3,2,4,3,2,19,18,17,16,41,149,148,147,146,145,225,224,223,222,221,220,
%T A025509 219,218,217,377,376,375,577,576,575,574,573,572,571,570,569,568,567,566,
%U A025509 565,564,563,7697,7696,7695,7694,7693,7692,7691,7690,7689,7688,7687,7686
%N A025509 Least k>1 such that reverse of first n terms of A006928 repeats beginning at k-th term.
%K A025509 nonn
%O A025509 1,1
%A A025509 dww
%I A007456
%S A007456 0,1,3,2,4,3,4,3,5,4,5,4,5,4,5,4,6,5,6,5,6,5,6,5,6,5,6,5,6,5,6,5,7,6,7,
%T A007456 6,7,6,7,6,7,6,7,6,7,6,7,6,7,6,7,6,7,6,7,6,7,6,7,6,7,6,7,6,8,7,8,7,8,7,
%U A007456 8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8
%N A007456 Days required to spread gossip to n people.
%C A007456 On first day, each gossip has his own tidbit. On each successive day, disjoint pairs of gossips may share tidbits (over the phone). After a(n) days, all gossips have all tidbits.
%D A007456 Fan, C. Kenneth, Bjorn Poonen and George Poonen, How to spread rumors fast. Mathematics Magazine 70 (February, 1997), pp. 40-42.
%H A007456 Source
%F A007456 a(n) = [ log_2 n+1 ] + (n mod 2) + 1 (n > 1)
%K A007456 nonn,nice,easy
%O A007456 1,3
%A A007456 Alex Graesser (AlexG@sni.co.za)
%E A007456 More terms from dww
%I A052938
%S A052938 1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,10,9,11,10,12,11,13,12,14,13,15,14,16,
%T A052938 15,17,16,18,17,19,18,20,19,21,20,22,21,23,22,24,23,25,24,26,25,27,26,
%U A052938 28,27,29,28,30,29,31,30,32,31,33,32,34,33,35,34,36,35,37,36,38,37,39
%N A052938 A simple regular expression.
%H A052938 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 929
%F A052938 G.f.: -(-2*x+2*x^2-1)/(-1+x)/(-1+x^2)
%F A052938 Recurrence: {a(0)=1,a(2)=2,a(1)=3,a(n)+a(n+1)-n-4}
%F A052938 3/4*(-1)^(1-n)+1/2*n+7/4
%p A052938 spec:= [S,{S=Prod(Union(Sequence(Z),Z,Z),Sequence(Prod(Z,Z)))},unlabelled ]: seq(combstruct[count ](spec,size=n),n=0..20);
%K A052938 easy,nonn
%O A052938 0,2
%A A052938 encyclopedia@pommard.inria.fr, Jan 25 2000
%E A052938 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 06 2000
%I A025532
%S A025532 0,0,1,1,3,2,4,3,5,5,7,5,8,7,7,6,10,8,11,9,10,10,12,9,12,12,12,12,15,13,16
%N A025532 a(n) = sum of exponents in prime factorization of LCM{C(n,0), C(n,1), ..., C(n,n)}.
%K A025532 nonn
%O A025532 0,5
%A A025532 Clark Kimberling (ck6@cedar.evansville.edu)
%I A026923
%S A026923 0,0,1,0,1,1,3,2,4,3,6,5,8,7,11,9,13,12,17,15,20,18,24,22,28,
%T A026923 26,33,30,37,35,43,40,48,45,54,51,60,57,67,63,73,70,81,77,88,
%U A026923 84,96,92,104,100,113,108,121,117,131
%N A026923 Partitions of n into an odd number of parts, the greatest being 3; also, a(n+5) = number of partitions of n+2 into an even number of parts, each <=3.
%K A026923 nonn
%O A026923 1,7
%A A026923 Clark Kimberling, ck6@cedar.evansville.edu
%I A058708
%S A058708 1,0,1,1,3,2,4,3,6,5,9,8,14,12,19,18,27,26,37,36,52,50,69,68,93
%N A058708 McKay-Thompson series of class 54A for Monster.
%D A058708 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058708 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058708 nonn
%O A058708 -1,5
%A A058708 njas, Nov 27, 2000
%I A024856
%S A024856 0,1,3,2,4,3,6,9,13,12,17,16,21,19,25,32,40,39,47,45,54,52,62,60,71,82,94,
%T A024856 92,105,103,117,115,130,127,142,139,155,172,190,188,207,205,224,221,241,238,
%U A024856 259,256,278,275,298,322,346,343,368,365,391,388,415,412,440,437,466
%N A024856 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (natural numbers), t = A023532.
%K A024856 nonn
%O A024856 2,3
%A A024856 Clark Kimberling (ck6@cedar.evansville.edu)
%I A023869
%S A023869 0,1,3,2,4,3,6,10,15,14,19,18,24,23,30,29,37,36,45,54,64,63,74,73,85,84,97,
%T A023869 96,110,109,124,123,139,138,154,152,169,167,185,204,224,223,244,243,265,264,
%U A023869 287,286,310,309,334,333,359,358,385,383,410,408,436,434,463,461,491,489
%N A023869 a(n) = s(1)t(n)+s(2)t(n-1)+...+s(k)t(n+1-k), where k=[ (n+1)/2 ], s = (natural numbers), t is A023534.
%K A023869 nonn
%O A023869 1,3
%A A023869 Clark Kimberling (ck6@cedar.evansville.edu)
%I A024596
%S A024596 0,1,3,2,4,3,6,11,19,18,30,29,48,45,74,66,108,87,142,230,373,372,603,600,
%T A024596 972,964,1561,1540,2493,2438,3946,3802,6153,5776,9346,8358,13525,10939,
%U A024596 17701,28642,46345,46332,74968,74934,121247,121158,196039,195806
%N A024596 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = [ (n+1)/2 ], s = (F(2), F(3), ...), t = A023534.
%K A024596 nonn
%O A024596 1,3
%A A024596 Clark Kimberling (ck6@cedar.evansville.edu)
%I A065375
%S A065375 1,1,3,2,4,3,7,5,17,12,24,17,41,29,99,70,140,99,239,169,577,408,816,
%T A065375 577,1393,985,3363,2378,8119,5741,19601,13860,47321,33461,114243,80782,
%U A065375 275807,195025
%N A065375 The next smallest pair of numbers is taken so that a(2n-1)/a(2n) converges to sqrt(2).
%C A065375 "Next smallest" means that a(2n-1)+a(2n) is the smallest value that is greater than the previous pair.
%e A065375 abs(1/1-sqr(2)) > abs(3/2-sqr(2)) > abs(4/3-sqr(2)) > abs(7/5-sqr(2)) > abs(17/12-sqr(2)) > ... > 0
%K A065375 nonn
%O A065375 1,3
%A A065375 Bodo Zinser (BodoZinser@Compuserve.com), Nov 22 2001
%I A061721
%S A061721 0,0,1,3,2,4,3,10,4,8,5,15,6,12,7,26
%N A061721 Number of zeros in the character table of the dihedral group with 2n elements .
%C A061721 For odd n a(n) = (n-1)/2
%e A061721 a(3) = 1 because the group is isomorphic to S_3 and the table is : 1, 1, 1 1,-1, 1 2, 0,-1
%Y A061721 A060762.
%K A061721 nonn
%O A061721 1,4
%A A061721 Ahmed Fares (ahmedfares@my-deja.com), Jun 20 2001
%I A062964
%S A062964 3,2,4,3,15,6,10,8,8,8,5,10,3,0,8,13,3,1,3,1,9,8,10,2,14,0,3,7,0,7,3,4,
%T A062964 4,10,4,0,9,3,8,2,2,2,9,9,15,3,1,13,0,0,8,2,14,15,10,9,8,14,12,4,14,6,
%U A062964 12,8,9,4,5,2,8,2,1,14,6,3,8,13,0,1,3,7,7,11,14,5,4,6,6,12,15,3,4,14,9
%N A062964 Pi in hexadecimal.
%H A062964 More digits
%H A062964 Steven Finch, The Miraculous Bailey-Borwein-Plouffe Pi Algorithm
%F A062964 a(n) =8*A004601(4n)+4*A004601(4n+1)+2*A004601(4n+2)+1*A004601(4n+3)
%Y A062964 Cf. A000796, A004601.
%K A062964 base,easy,nonn
%O A062964 0,1
%A A062964 Robert Lozyniak (11@onna.com), Jul 22 2001
%E A062964 More terms from Henry Bottomley (se16@btinternet.com), Jul 24 2001
%I A010270
%S A010270 3,2,4,3,19,1,36,2,3,3,1,4,4,3,11,1,1,3,1,2,97,1,3,1,1,
%T A010270 1,3,1,16,2,3,7,1,6,2,2,1,4,3,1,6,1,3,11,1,1,1,15,1,2,4,
%U A010270 4,1,4,3,1,1,1,4,2,2,3,1,2,4,3,1,1,4,1,1,2,6,1,22,1,4,4
%N A010270 Continued fraction for cube root of 41.
%K A010270 nonn,cofr
%O A010270 0,1
%A A010270 njas
%I A023630
%S A023630 3,2,4,4,2,2,4,2,2,4,4,4,2,2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,2,2,
%T A023630 4,2,2,4,4,4,2,2,4,4,4,2,2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,4,4,2,
%U A023630 2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,2,2,4,2,2,4,4,4,2,2,4,2,2,4,2
%N A023630 a(n) = s(2n) - s(2n-1), where s( ) is sequence A023629.
%K A023630 nonn
%O A023630 1,1
%A A023630 Clark Kimberling (ck6@cedar.evansville.edu)
%I A061901
%S A061901 1,1,1,1,1,1,1,1,1,1,3,2,4,4,4,4,4,4,4,4,3,4,2,4,4,4,4,4,4,4,3,4,4,2,4,4,
%T A061901 4,4,4,4,3,4,4,4,2,4,4,4,4,4,3,4,4,4,4,2,4,4,4,4,3,4,4,4,4,4,2,4,4,4,3,
%U A061901 4,4,4,4,4,4,2,4,4,3,4,4,4,4,4,4,4,2,4,3,4,4,4,4,4,4,4,4,2,4,6,11,11
%N A061901 Number of distinct numbers that can be formed from the digits of n.
%e A061901 The digits of 13 can form 1, 3, 13, and 31. The digits of 201 can form 0, 1, 2, 10, 12, 20, 21, 102, 120, 210, and 201. Thus a(13) = 4, a(201) = 11.
%Y A061901 Cf. A061487, A061497, A061499, A061500.
%K A061901 nonn,base
%O A061901 0,11
%A A061901 Asher Auel (asher.auel@reed.edu), 15 May 2001
%I A059851
%S A059851 1,1,3,2,4,4,6,4,7,7,9,7,9,9,13,10,12,12,14,12,16,16,18,14,17,17,21,19,
%T A059851 21,21,23,19,23,23,27,24,26,26,30,26,28,28,30,28,34,34,36,30,33,33,37,
%U A059851 35,37,37,41,37,41,41,43,39,41,41,47,42,46,46,48,46,50,50,52,46,48,48
%N A059851 a(n) = n - [n/2] + [n/3] - [n/4] + ... (this is a finite sum).
%C A059851 as n goes to infinity we have the asymptotic formula: a(n) ~ n * ln(2)
%e A059851 a(5) = 4 because : [5] - [5/2] + [5/3] - [5/4] + [5/5] - [5/6] + ... = 5 - 2 + 1 - 1 + 1 - 0 + 0 - 0 + ... = 4
%p A059851 for n from 1 to 200 do printf(`%d,`, sum((-1)^(i+1)*floor(n/i), i=1..n)) od:
%K A059851 nonn,easy
%O A059851 1,3
%A A059851 Avi Peretz (njk@netvision.net.il), Feb 27 2001
%E A059851 More terms from James A. Sellers (sellersj@math.psu.edu) and Larry Reeves (larryr@acm.org), Feb 27 2001.
%I A047993
%S A047993 1,0,1,1,1,1,3,2,4,4,6,7,11,11,16,19,25,29,40,45,60,70,89,105,134,156,
%T A047993 196,232,285,336,414,485,591,696,839,987,1187,1389,1661,1946,2311,2702,
%U A047993 3201,3731,4400,5126,6018,6997,8195,9502,11093,12849,14949,17281,20062
%N A047993 Number of balanced partitions of n: first element is equal to number of elements.
%C A047993 Useful in the creation of plane partitions with C3 or C3v symmetry
%C A047993 The function T[m,a,b] used here gives the partitions of n whose Ferrers plot fits within an a X b box.
%e A047993 {5,4,1,1,1} is a balanced partition of 12 because the first element is 5, and the length is 5.
%t A047993 Table[ Count[Partitions[n],par_List/;First[par]===Length[par]],{n,12}] or recur: Sum[T[n-(2m-1),m-1,m-1],{m,Ceiling[Sqrt[n]],Floor[(n+1)/2]}] with T[m_,a_,b_]/;b < a:=T[m,b,a];T[m_,a_,b_]/;m > a*b:=0; T[m_,a_,b_]/;(2m > a*b):=T[a*b-m,a,b]; T[m_,1,b_]:=If[b < m,0,1];T[0,_,_]:=1; T[m_,a_,b_]:=T[m,a,b]=Sum[T[m-a*i,a-1,b-i],{i,0,Floor[m/a]}];
%K A047993 easy,nice,nonn
%O A047993 1,7
%A A047993 Wouter Meeussen (w.meeussen.vdmcc@vandemoortele.be)
%I A033177
%S A033177 0,1,1,1,3,2,4,4,6,7,20,3,28,9,21,12,14,18,52,26,69,17,48,16,160,168
%N A033177 Number of distinct distances between n electrons in minimal energy configuration on a sphere.
%C A033177 Most of the higher terms of this sequence just refer to the best configurations known today.
%H A033177 K. S. Brown, Points on Sphere
%H A033177 Table of best configurations presently known
%K A033177 more,nonn
%O A033177 1,5
%A A033177 Kevin Brown (ksbrown@seanet.com)
%E A033177 How is this defined if the optimal configuration is not unique? - NJAS
%I A021759
%S A021759 0,0,1,3,2,4,5,0,3,3,1,1,2,5,8,2,7,8,1,4,5,6,9,5,3,6,4,2,3,8,4,1,0,
%T A021759 5,9,6,0,2,6,4,9,0,0,6,6,2,2,5,1,6,5,5,6,2,9,1,3,9,0,7,2,8,4,7,6,8,
%U A021759 2,1,1,9,2,0,5,2,9,8,0,1,3,2,4,5,0,3,3,1,1,2,5,8,2,7,8,1,4,5,6,9,5
%N A021759 Decimal expansion of 1/755.
%K A021759 nonn,cons
%O A021759 0,4
%A A021759 njas
%I A020814
%S A020814 1,3,2,4,5,3,2,3,5,7,0,6,5,0,4,3,8,0,6,5,3,2,7,5,1,7,2,0,5,1,6,8,6,
%T A020814 2,6,4,1,7,9,3,3,6,7,4,9,2,0,3,9,1,8,9,4,9,7,9,4,7,1,2,7,5,2,7,3,1,
%U A020814 7,6,7,0,5,1,8,2,3,1,1,9,4,9,4,6,5,9,3,4,1,2,6,4,4,5,8,7,2,8,4,9,0
%N A020814 Decimal expansion of 1/sqrt(57).
%K A020814 nonn,cons
%O A020814 0,2
%A A020814 njas
%I A063201
%S A063201 0,1,3,2,4,5,5,6,8,7,9,10,10,11,13,12,14,15,15,16,18,17,19,20,20,
%T A063201 21,23,22,24,25,25,26,28,27,29,30,30,31,33,32,34,35,35,36,38,37,
%U A063201 39,40,40,41
%N A063201 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 18 ).
%H A063201 William A. Stein (was@math.berkeley.edu), Dimensions of the spaces S_k^{new}(Gamma_0(N))
%H A063201 William A. Stein (was@math.berkeley.edu), The modular forms database
%K A063201 nonn
%O A063201 1,3
%A A063201 njas, Jul 10 2001
%I A039858
%S A039858 0,1,1,1,1,3,2,4,5,5,8,10,11,17,18,25,29,38,45,56,69,84,101,127,148,
%T A039858 183,219,264,314,382,447,542,639,760,902,1067,1256,1489,1742,2060,2402,
%U A039858 2832,3298,3862,4507
%N A039858 Partitions satisfying cn(0,5) = cn(1,5) + cn(4,5).
%C A039858 For a given partition cn(i,n) means the number of its parts equal to i modulo n.
%C A039858 Short: 0 = 1 + 4 (AApEZ).
%K A039858 nonn,part
%O A039858 1,6
%A A039858 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A035558
%S A035558 1,0,1,1,1,3,2,4,5,5,12,8,16,18,20,37,31,50,59,64,108,96,146,169,189,286,
%T A035558 275,388,452,504,725,718,977,1129,1270,1733,1778,2325,2688,3022,3991,
%U A035558 4165,5319,6116,6896,8837,9380,11703,13433,15130,18979,20343,24966,28543
%N A035558 Partitions of n with equal number of parts congruent to each of 1 and 4 (mod 5)
%K A035558 nonn,part
%O A035558 0,6
%A A035558 Olivier Gerard (ogerard@ext.jussieu.fr)
%E A035558 More terms from dww
%I A035044
%S A035044 0,1,3,2,4,5,6,7,8,9,10,11,13,12,14,15,16,17,18,19,30,31,33,32,34,
%T A035044 35,36,37,38,39,20,21,23,22,24,25,26,27,28,29,40,41,43,42,44,45,46,
%U A035044 47,48,49,50,51,53,52,54,55,56,57,58,59,60,61,63,62,64,65,66,67,68
%N A035044 Exchange 2 and 3!.
%K A035044 nonn,base,easy
%O A035044 0,3
%A A035044 njas
%I A036812
%S A036812 0,1,1,1,1,3,2,4,5,6,9,11,12,19,22,30,34,44,54,69,85,103,122,155,184,
%T A036812 227,271,325,388,473,557,674,788,939,1113,1319,1554,1830,2137,2523,
%U A036812 2943,3467,4020,4688,5454
%N A036812 Partitions satisfying (cn(0,5) = 0 and cn(1,5) <= cn(2,5) and cn(1,5) <= cn(3,5) and cn(4,5) <= cn(2,5) and cn(4,5) <= cn(3,5)).
%C A036812 For a given partition cn(i,n) means the number of its parts equal to i modulo n.
%C A036812 Short: (0:=0 and 1<=2 and 1<=3 and 4<=2 and 4<=3).
%K A036812 nonn,part
%O A036812 1,6
%A A036812 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A039906
%S A039906 0,1,1,1,1,3,2,4,5,6,9,11,12,19,22,30,34,44,54,69,86,103,123,157,187,
%T A039906 232,276,331,399,490,579,698,820,983,1179,1398,1646,1944,2288,2730,
%U A039906 3190,3758,4374,5136,6043
%N A039906 Partitions satisfying cn(0,5) + cn(1,5) + cn(4,5) <= cn(2,5) and cn(0,5) + cn(1,5) + cn(4,5) <= cn(3,5).
%C A039906 For a given partition cn(i,n) means the number of its parts equal to i modulo n.
%C A039906 Short: 0 + 1 + 4 <= 2 and 0 + 1 + 4 <= 3 (ZAApBB).
%K A039906 nonn,part
%O A039906 1,6
%A A039906 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A056011
%S A056011 1,3,2,4,5,6,10,9,8,7,11,12,13,14,15,21,20,19,18,17,16,22,23,24,25,26,
%T A056011 27,28,36,35,34,33,32,31,30,29,37,38,39,40,41,42,43,44,45,55,54,53,52,
%U A056011 51,50,49,48,47,46,56,57,58,59,60,61,62,63,64,65,66
%N A056011 Unique triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are increasing; and (4) even-numbered rows are decreasing.
%C A056011 Self-inverse permutation of the natural numbers.
%H A056011 Index entries for sequences that are permutations of the natural numbers
%e A056011 1; 3,2; 4,5,6; 10,9,8,7; 11,12,13,14,15; ...
%K A056011 nonn,tabl,nice
%O A056011 1,2
%A A056011 Clark Kimberling, ck6@evansville.edu, Aug 01 2000
%I A039882
%S A039882 0,1,1,1,1,3,2,4,5,7,9,12,13,20,26,33,38,50,61,83,99,122,145,185,230,
%T A039882 280,335,406,490,616,725,877,1034,1250,1518,1805,2127,2524,2993,3610,
%U A039882 4229,4991,5831,6897,8185
%N A039882 Partitions satisfying cn(0,5) + cn(1,5) <= cn(2,5) and cn(0,5) + cn(1,5) <= cn(3,5) and cn(0,5) + cn(4,5) <= cn(2,5) and cn(0,5) + cn(4,5) <= cn(3,5).
%C A039882 For a given partition cn(i,n) means the number of its parts equal to i modulo n.
%C A039882 Short: 0 + 1 <= 2 and 0 + 1 <= 3 and 0 + 4 <= 2 and 0 + 4 <= 3 (ZAABB).
%K A039882 nonn,part
%O A039882 1,6
%A A039882 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A001612 M0974 N0364
%S A001612 3,2,4,5,8,12,19,30,48,77,124,200,323,522,844,1365,2208,3572,5779,9350,
%T A001612 15128,24477,39604,64080,103683,167762,271444,439205,710648,1149852,
%U A001612 1860499,3010350,4870848,7881197,12752044,20633240,33385283,54018522
%N A001612 a(n) = a(n-1) + a(n-2) - 1.
%D A001612 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
%F A001612 G.f.: (3-4*x)/((1-x)*(1-x-x^2)). a(n)=a(n-1)+a(n-2)-1.
%o A001612 (PARI) a(n)=fibonacci(n+1)+fibonacci(n-1)+1
%Y A001612 Lucas sequence A000032 + 1.
%K A001612 nonn
%O A001612 0,1
%A A001612 njas
%E A001612 Additional comments from Michael Somos, Jun 01 2000.
%I A059320
%S A059320 0,0,3,2,4,5,10,4,13,4,14,13,16,18,21,10,22,23,22,14,25,22,36,25,42,24,
%T A059320 41,40,40,35,48,20,51,34,50,51,50,30,63,24,64,35,56,52,65,64,74,53,76,
%U A059320 78,73,58,78,71,94,78,99,64,92,79,84,84,95,42,96,89,90,80,103,82,122
%N A059320 Number of 0's in row n of Pascal's rhombus mod 2.
%D A059320 J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 321-236.
%Y A059320 Cf. A059317, A059318, A059319, A007318.
%K A059320 easy,nonn
%O A059320 0,3
%A A059320 njas, Jan 26 2001
%E A059320 More terms from Larry Reeves (larryr@acm.org), Jan 30 2001
%I A049831
%S A049831 0,0,1,1,3,2,4,6,5,6,8,8,11,10,9,12,12,15,15,14,19,16,19,18,19,22,21,
%T A049831 23,26,25,27,24,27,32,27,30,33,31,33,30,35,38,35,38,40,38,44,39,44,46,
%U A049831 43,44,47,45,53,46,49,52,50,56,54,54,57,56
%N A049831 a(n)=MAX{T(n,k): k=1,2,...,n}, array T as in A049828.
%K A049831 nonn
%O A049831 1,5
%A A049831 Clark Kimberling, ck6@cedar.evansville.edu
%I A059417
%S A059417 1,1,3,2,4,6,5,25,27,26,676,678,677,458329,458331,458330,210066388900,
%T A059417 210066388902,210066388901,44127887745906175987801,
%U A059417 44127887745906175987803,44127887745906175987802
%N A059417 Start with 1; square; add 2; subtract 1; repeat.
%D A059417 Seen on a quiz.
%K A059417 easy,huge,nonn
%O A059417 0,3
%A A059417 Jonathan Scharff (jonscharff@home.com), Jan 30 2001
%E A059417 More terms from Larry Reeves (larryr@acm.org), Jan 31 2001. The next term has 46 digits.
%I A014685
%S A014685 1,3,2,4,6,6,6,8,9,10,12,12,12,14,15,16,18,18,18,20,21,22,24,24,25,26,
%T A014685 27,28,28,30,32,32,33,34,35,36,36,38,39,40,42,42,42,44,45,46,48,48,49,
%U A014685 50,51,52,52,54,55,56,57,58,60,60,60,62,63,64,65,66,68,68,69,70,70,72
%N A014685 In sequence of positive integers add 1 to 1st prime and subtract 1 from 2nd prime; add 1 to 3rd prime and subtract 1 from 4th prime and so on.
%K A014685 nonn,easy
%O A014685 1,2
%A A014685 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A014685 More terms from Andrew J. Gacek (andrew@dgi.net)
%I A021312
%S A021312 0,0,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,
%T A021312 2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,
%U A021312 7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3,2,4,6,7,5,3
%N A021312 Decimal expansion of 1/308.
%K A021312 nonn,cons
%O A021312 0,3
%A A021312 njas
%I A019653
%S A019653 3,2,4,6,9,8,0,4,0,5,9,1,3,6,4,2,0,7,1,7,1,2,9,1,4,5,2,9,0,0,8,5,3,
%T A019653 5,9,0,5,9,9,7,6,7,5,0,5,8,4,9,6,1,8,5,3,9,8,9,0,8,6,0,0,0,1,2,7,4,
%U A019653 6,6,1,7,9,6,9,2,9,2,1,7,4,3,1,6,0,4,8,8,4,0,6,3,0,8,5,2,2,8,7,6,9
%N A019653 Decimal expansion of sqrt(Pi*E)/9.
%K A019653 nonn,cons
%O A019653 0,1
%A A019653 njas
%I A039915
%S A039915 3,2,4,6,10,12,16,18,22,28,30,36,40,42,46,52,58,60,66,70,72,78,82,88,
%T A039915 96,100,102,106,108,112,126,130,136,138,148,150,156,162,166,172,178,
%U A039915 180,190,192,196,198,210,222,226,228
%N A039915 Smallest k such that k(p-1)-1 is divisible by p; p=nth prime.
%e A039915 a(1)=3 because 3(2-1)-1=2 is divisible by 2.
%Y A039915 A039678.
%K A039915 base,nice,nonn
%O A039915 1,1
%A A039915 Felice Russo (felice.russo@katamail.com)
%I A060006
%S A060006 1,3,2,4,7,1,7,9,5,7,2,4,4,7,4,6,0,2,5,9,6,0,9,0,8,8,5,4
%N A060006 Decimal expansion of v_3, where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1.
%H A060006 F. Rothelius, Formulae
%e A060006 v_3 = 1.324717957244746025960908854...
%Y A060006 v_2 = A001622.
%K A060006 base,cons,nice,nonn
%O A060006 1,2
%A A060006 Fabian Rothelius (fabian.rothelius@telia.com), Mar 14 2001
%I A054086
%S A054086 1,3,2,4,7,5,6,11,8,9,14,10,12,18,13,15,22,16,17,26,19,20,29,21,23,33,
%T A054086 24,25,37,27,28,41,30,31,44,32,34,48,35,36,52,38,39,55,40,42,59,43,45,
%U A054086 63,46,47,67,49,50,70,51,53,74,54,56,78,57
%N A054086 For k >= 1, let p(k)=least h in N not already an a(i), q(k)=p(k)+k, a(3k-1)=q(k), a(3k)=p(k), a(3k+1)=least h in N not already an a(i).
%Y A054086 The trisection (3, 7, 11, 14, ...) is A003512, with n-th term [n(2+sqrt(3)]; the complement of this is A003511, with n-th term [n(1+sqr(3))/2].
%Y A054086 A054087, A054088.
%K A054086 nonn,eigen
%O A054086 1,2
%A A054086 Clark Kimberling, ck6@cedar.evansville.edu
%I A014193
%S A014193 3,2,4,7,10,14,16,19,23,30,30,37,40,44,48,53,58,61,66,71,
%T A014193 74,80,82,89,97,102,103,107,108,112,126,131,138,140,150,
%U A014193 151,156,164,168,173,178,180,190,193,197,200,210,223,227
%N A014193 n-th prime + Mobius(n).
%p A014193 with(numtheory): f:=n->ithprime(n)+mobius(n); [ seq(f(i),i=1..80) ];
%K A014193 nonn
%O A014193 1,1
%A A014193 njas
%I A019916
%S A019916 3,2,4,9,1,9,6,9,6,2,3,2,9,0,6,3,2,6,1,5,5,8,7,1,4,1,2,2,1,5,1,3,4,
%T A019916 4,6,4,9,5,4,9,0,3,4,7,1,5,2,1,4,7,5,1,0,0,3,0,7,8,0,4,7,1,9,1,3,6,
%U A019916 6,7,2,9,0,0,9,6,0,7,4,4,9,4,8,3,2,2,6,8,7,7,3,5,4,4,6,9,6,5,0,5,0
%N A019916 Decimal expansion of tangent of 18 degrees.
%K A019916 nonn,cons
%O A019916 0,1
%A A019916 njas
%I A033820
%S A033820 1,1,3,2,4,10,5,9,15,35,14,24,36,56,126,42,70,100,140,210,462,132,216,
%T A033820 300,400,540,792,1716
%N A033820 If a(k,j) is our sequence read as Pascal triangle by row k, then f(n,k)=2^{n-2(k-2)}sum(a(k-2,j)*binomial(n+2*(k-2-j),2*(k-2-j)),j=0..k-2) is the number of n-long k-ary strings (k >= 2) which avoid a rising triple (pattern 123) or any other given 3-letter permutation pattern.
%C A033820 {1},{1,3},{2,4,10},{5,9,15,35},{14,24,36,56,126},{42,70,100,140,210,462}, {132,216,300,400,540,792,1716} a(k,0)=binomial(2*k,k)/(k+1), the k-th Catalan number a(k,k)=binomial(2*(k+1),k+1)/2, half the (k+1)-st central binomial coefficient sum of entries in row k (over j) = 2^{2*(k-1)}
%D A033820 Alexander Burstein, Enumeration of words with forbidden patterns, Ph.D. thesis, University of Pennsylvania, 1998.
%H A033820 Thesis above
%F A033820 a(k,j)=sum(C(k-i)D(i),i=0..j), C(i)=binomial(2*i,i)/(i+1), D(i)=binomial(2*i,i), G.f.: sum(sum(a(k,j)*x^k*y^j,j=0..k),k=0..infinity)=2*x^2/(1-x+sqrt((1-x)*(1-x*y)))
%Y A033820 Cf. A000108, A000984, A000302.
%K A033820 nonn,tabl
%O A033820 0,3
%A A033820 Alexander Burstein (alexb@math.upenn.edu)
%I A019321
%S A019321 3,2,4,13,10,121,7,1093,82,757,61,88573,73,797161,547,4561,
%T A019321 6562,64570081,703,581130733,5905,368089,44287,47071589413,
%U A019321 6481,3501192601,398581,387440173,478297,34315188682441
%N A019321 Cyclotomic polynomials at x=3.
%H A019321 Index entries for cyclotomic polynomials, values at X
%p A019321 with(numtheory,cyclotomic);f:=n->subs(x=3,cyclotomic(n,x));seq(f(i),i=0..64);
%K A019321 nonn
%O A019321 0,1
%A A019321 sp
%I A019116
%S A019116 0,1,3,2,4,24,40,93,297,575,1321,3793,9023,22074,59286,150414,
%T A019116 378079,988794
%N A019116 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite DOH = Dodecasil 1H [ Si34O68 ] . q R.
%D A019116 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019116 G. Thimm, Cycle sequences of crystal structures
%K A019116 nonn
%O A019116 3,3
%A A019116 Georg Thimm (mgeorg@ntu.edu.sg)
%I A021887
%S A021887 0,0,1,1,3,2,5,0,2,8,3,1,2,5,7,0,7,8,1,4,2,6,9,5,3,5,6,7,3,8,3,9,1,
%T A021887 8,4,5,9,7,9,6,1,4,9,4,9,0,3,7,3,7,2,5,9,3,4,3,1,4,8,3,5,7,8,7,0,8,
%U A021887 9,4,6,7,7,2,3,6,6,9,3,0,9,1,7,3,2,7,2,9,3,3,1,8,2,3,3,2,9,5,5,8,3
%N A021887 Decimal expansion of 1/883.
%K A021887 nonn,cons
%O A021887 0,5
%A A021887 njas
%I A051543
%S A051543 3,2,5,1,7,2,3,1,11,1,13,1,1,2,17,1,19,1,1,1,23,1,5,1,3,1,29,1,31,2,1,
%T A051543 1,1,1,37,1,1,1,41,1,43,1,1,1,47,1,7,1,1,1,53,1,1,1,1,1,59,1,61,1,1,2,
%U A051543 1,1,67,1,1,1,71,1,73,1,1,1,1,1,79,1,3,1,83,1,1,1,1,1,89,1,1,1,1,1,1,1
%N A051543 Quotients of consecutive values of lcm of first n triangular numbers (A000217).
%F A051543 a(n) = A051537(n+1)/A051537(n)
%e A051543 a(5) = A051537(6)/A051537(5) = 210/30 = 7
%Y A051543 A051537.
%K A051543 easy,nonn
%O A051543 1,1
%A A051543 Asher Auel (asher.auel@reed.edu)
%E A051543 Corrected and extended by James A. Sellers (sellersj@math.psu.edu)
%I A056008
%S A056008 3,2,5,1,9,4,17,16,33,17,65,68,129,89,257,356,513,697,1025,1337,2049,
%T A056008 2449,4097,4001,8193,4417,16385,17668,32769,24329,65537,4633,131073,
%U A056008 18532,262145,74128,524289,296512,1048577,1186048,2097153,1778369
%N A056008 Difference between (smallest square strictly greater than 2^n) and 2^n.
%e A056008 a(5)=6^2-2^5=4; a(6)=9^2-2^6=17
%Y A056008 Cf. A051204.
%K A056008 nonn
%O A056008 0,1
%A A056008 Henry Bottomley (se16@btinternet.com), Jul 24 2000
%I A057034
%S A057034 3,2,5,2,5,4,7,4,7,4,23,2,19,2,97,2,19,64,17,62,19,4,17,58,23,64,157,
%T A057034 122,47,106,47,4,29,146,31,64,29,8,71,8,43,32,31,128,67,122,41,2,37,
%U A057034 146,137,122,191,142,59,128,71,284,109,274,101,218,97,32,83,158,53,166
%N A057034 Difference between n!! and the first prime before n!! - 1.
%C A057034 Analogous to the lesser Fortunate numbers but unlike them, all entries are not prime. Must odd entries are powers of two and all even entries are prime.
%t A057034 PrevPrime[ n_Integer ]:= (k=n-1; While[ !PrimeQ[ k ],k-- ]; Return[ k ]); f[ n_Integer ] := (p = n!! - 1; q = PrevPrime[ p ]; Return[ p - q + 1 ]); Table[ f[ n ], {n, 4, 75} ]
%K A057034 nonn
%O A057034 4,1
%A A057034 Robert G. Wilson v (RGWv@kspaint.com), Sep 09 2000
%I A023513
%S A023513 3,2,5,2,5,9,5,7,5,10,2,21,12,13,5,5,10,33,19,5,39,7,12,10,9,22,15,
%T A023513 5,18,24,2,16,28,14,10,21,81,43,12,34,10,22,5,99,16,7,55,9,24,30,18,
%U A023513 10,13,12,48,16,10,19,141,52,73,12,20,18,159,58,85,15,34,14,64,10
%N A023513 Sum of distinct prime divisors of p(n) + 1.
%K A023513 nonn
%O A023513 1,1
%A A023513 Clark Kimberling (ck6@cedar.evansville.edu)
%I A046524
%S A046524 1,3,2,5,2,7,2,8,3,8,2,12,2,9,4,13,2,14,2,16
%N A046524 Coverings of Klein bottle with n lists.
%D A046524 A. D. Mednykh, On the number of subgroups in the fundamental group of a closed surface. Commun. in Algebra, 16, No 10 (1988), 2137-2148.
%F A046524 a(n)=d(n) (the number of divisors) for odd n. An easy formula for even n.
%Y A046524 Cf. A027842, A027844.
%K A046524 nonn,easy,nice
%O A046524 1,2
%A A046524 V. A. Liskovets (liskov@im.bas-net.by)
%I A046227
%S A046227 3,2,5,2,11,2,7,2,9,9,2,20,27,2,11,2,67,2,19,13,2,147,2,105,2,51,2,15,
%T A046227 2,126,78,33,17,2
%N A046227 First numerator and then denominator of elements to right of central elements of 1/2-Pascal triangle (by row), excluding 1's.
%e A046227 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...
%Y A046227 Cf. A046213.
%K A046227 tabl,nonn
%O A046227 1,1
%A A046227 Mohammad K. Azarian, ma3@cedar.evansville.edu
%I A057958
%S A057958 1,3,2,5,3,5,2,7,3,6,3,8,2,5,5,10,3,8,3,10,4,7,3,11,5,5,6,9,4,11,4,12,
%T A057958 5,8,6,12,3,7,7,13,4,11,3,11,9,6,5,17,7,10,6,9,4,13,8,13,7,9,3,17,3,8,
%U A057958 6,14,7,12,4,12,6,11,2,16,5,8,10,11,7,15,4,18,9,8,5,18,7,6,8,16,4,19,5
%N A057958 Number of prime factors of 3^n - 1 (counted with multiplicity).
%H A057958 S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
%H A057958 S. S. Wagstaff, Jr., The Cunningham Project
%Y A057958 Cf. A057951-A057958, A046051.
%K A057958 nonn
%O A057958 0,2
%A A057958 Patrick De Geest (pdg@worldofnumbers.com), Nov 2000.
%I A057953
%S A057953 1,3,2,5,3,6,4,7,3,7,4,10,4,8,6,10,5,9,4,13,7,9,4,14,7,8,6,14,6,13,3,
%T A057953 13,8,11,11,15,6,9,9,17,5,14,5,15,10,9,6,19,7,14,8,18,8,16,10,19,7,11,
%U A057953 6,24,5,8,10,16,8,17,6,20,9,22,7,21,7,13,14,17,10,16,8,23,10,12,5,24
%N A057953 Number of prime factors of 8^n - 1 (counted with multiplicity).
%H A057953 S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
%H A057953 S. S. Wagstaff, Jr., The Cunningham Project
%Y A057953 Cf. A057951-A057958, A046051.
%K A057953 nonn
%O A057953 1,2
%A A057953 Patrick De Geest (pdg@worldofnumbers.com), Nov 2000.
%I A030640
%S A030640 1,1,3,2,5,3,7,4,9,5,11,6,13,7,15,8,17,9,19,10,21,11,23,
%T A030640 12,25,13,27,14,29,15,31,16,33,17,35,18,37,19,39,20,41,
%U A030640 21,43,22,45,23,47,24,49,25,51,26,53,27,55,28,57,29,59
%V A030640 1,1,-3,-2,5,3,-7,-4,9,5,-11,-6,13,7,-15,-8,17,9,-19,-10,21,11,-23,
%W A030640 -12,25,13,-27,-14,29,15,-31,-16,33,17,-35,-18,37,19,-39,-20,41,
%X A030640 21,-43,-22,45,23,-47,-24,49,25,-51,-26,53,27,-55,-28,57,29,-59
%N A030640 Discriminant of lattice A_n of determinant n+1.
%D A030640 G. L. Watson, Integral Quadratic Forms, Camb., p. 2.
%D A030640 J. H. Conway, Sensual Quadratic Form, MAA, p. 4.
%F A030640 a(2n)=(-1)^n*(2*n+1), a(2n+1)=(-1)^n*(n+1). Or, a(n)=n, n odd; n/2, n even.
%F A030640 a(n+1)=(n+1)/gcd(n+1,2), g.f. (1+x+x^2)/(1-x^2)^2 (for the absolute values) - Len Smiley (smiley@math.uaa.alaska.edu).
%Y A030640 Essentially same as A026741. Partial sums give A001318.
%K A030640 sign,done,easy,nice
%O A030640 0,3
%A A030640 njas
%I A061366
%S A061366 1,3,2,5,3,7,4,9,5,11,6,13,7,15,8,17,9,19,10,21,11,23,12,25,13,27,14,
%T A061366 29,15,31,16,33,17,35,18,37,19,39,20,41,21,43,22,45,23,47,24,49,25,51,
%U A061366 26,53,27,55,28,57,29,59,30,61,31,63,32,65,33,67,34,69,35,71,36,73,37
%N A061366 Follow n with 2n+1.
%F A061366 a(n) = n+1 if n is even else a(n) = (n+1)/2
%Y A061366 Cf. A022998.
%K A061366 nonn,easy
%O A061366 1,2
%A A061366 Amarnath Murthy (amarnath_murthy@yahoo.com), May 02 2001
%E A061366 More terms from Larry Reeves (larryr@acm.org), May 07 2001
%I A026741
%S A026741 0,1,1,3,2,5,3,7,4,9,5,11,6,13,7,15,8,17,9,19,10,21,11,23,12,25,13,27,14,
%T A026741 29,15,31,16,33,17,35,18,37,19,39,20,41,21,43,22,45,23,47,24,49,25,51,26,
%U A026741 53,27,55,28,57,29,59,30,61,31,63,32,65,33,67,34,69,35,71,36,73,37,75,38
%N A026741 a(2n)=n, a(2n+1)=2n+1. Also numerator of n/((n+1)(n+2)) = A026741/A045896.
%C A026741 n / gcd(n,2). Cf. A026741, A051176, A060819, A060791, A060789 for n / gcd(n,k) for k=2..6.
%H A026741 M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
%F A026741 G.f.: (x^3+x^2+x)/(1-x^2)^2 - Len Smiley (smiley@math.uaa.alaska.edu), Apr 30 2001
%F A026741 a(n) = n * 2^((n mod 2) - 1) - Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Oct 16 2001
%Y A026741 Essentially same as A030640. Cf. A045896.
%K A026741 nonn,easy,nice,frac
%O A026741 0,4
%A A026741 J. Carl Bellinger (carlb@ctron.com)
%E A026741 More terms from dww; better description from Jud McCranie (jud.mccranie@mindspring.com)
%I A045766
%S A045766 0,1,1,3,2,5,3,8,5,10,6,13,7,15,9,19,10,22,11,25,13,27,14,31,16,33,19,
%T A045766 36,20,39,21,44,23,46,25,50,26,52,28,56,29,59,30,62,33,64,34,69,36,72,
%U A045766 38,75,39,79,41,83,43,85,44,89,45,91,48,97,50,100,51,103,53,106,54,111
%N A045766 Number of prime factors of double factorials n!! (A006882), with multiplicity.
%H A045766 Index entries for sequences related to factorial numbers
%e A045766 8!!=8*6*4*2=2^3*2*3*2^2*2=2^7*3. Eight prime factors, so a(8)=8.
%K A045766 nonn,easy
%O A045766 1,4
%A A045766 Jeff Burch (gburch@erols.com)
%I A061313
%S A061313 0,1,3,2,5,4,4,3,7,6,6,5,6,5,5,4,9,8,8,7,8,7,7,6,8,7,7,6,7,6,6,5,11,10,
%T A061313 10,9,10,9,9,8,10,9,9,8,9,8,8,7,10,9,9,8,9,8,8,7,9,8,8,7,8,7,7,6,13,12,
%U A061313 12,11,12,11,11,10,12,11,11,10,11,10,10,9,12,11,11,10,11,10,10,9,11,10
%N A061313 Minimal number of steps to get from 1 to n by (a) subtracting 1 and/or (b) multiplying by 2. A stopping problem: begin with n and at each stage if even divide by 2 or if odd add 1.
%F A061313 a(2n) = a(n)+1; a(2n+1) = a(n+1)+2; a(1) = 0.
%e A061313 a(2) = 1 since 2 = 1*2, a(3) = 3 since 3 = 1*2*2-1, a(11) = 6 since 11 = (1*2*2-1)*2*2-1.
%t A061313 f[n_] := Block[{c = 0, m = n},While[m != 1, If[ EvenQ[m], While[ EvenQ[m], m = m/2; c++ ], m++; c++ ]]; Return[c]]; Table[f[n], {n, 1, 100}]
%Y A061313 Cf. A056792.
%K A061313 easy,nonn
%O A061313 1,3
%A A061313 Henry Bottomley (se16@btinternet.com), Jun 06 2001
%I A053087
%S A053087 0,1,3,2,5,4,5,3,7,6,5,7,6,8,7,4,9,8,7,8,9,6,9,8,9,7,9,10,8,10,9,5,
%T A053087 11,10,9,10,8,10,11,9,10,11,7,11,10,11,9,11,10,11,12,8,11,10,12,11,
%U A053087 9,12,11,12,10,12,11,6,13,12,11,12,10,12,11,12,13,9,12,11,13,12,10
%N A053087 Lowest j for which 2^j kara n is defined.
%e A053087 a(6)=4 because 2^3 kara 6 is undefined, but 16 kara 6 = 3.
%Y A053087 Cf. A053405.
%K A053087 nonn
%O A053087 1,3
%A A053087 Robert Lozyniak (11@onna.com), Feb 26 2000
%E A053087 More terms from Antony M. Goddard, cat@animal.u-net.com or tony@noggin.lowtech.org, Oct 13, 2000.
%I A062327
%S A062327 1,3,2,5,4,6,2,7,3,12,2,10,4,6,8,9,4,9,2,20,4,6,2,14,9,12,4,10,4,24,2,
%T A062327 11,4,12,8,15,4,6,8,28,4,12,2,10,12,6,2,18,3,27,8,20,4,12,8,14,4,12,2,
%U A062327 40,4,6,6,13,16,12,2,20,4,24,2,21,4,12,18,10,4,24,2,36,5,12,2,20,16,6
%N A062327 Number of divisors of n over the Gaussian integers.
%C A062327 Divisors which are associates are identified (two Gaussian integers z1, z2 are associates if z1 = u * z2 where u is a unit, i.e. one of 1, i, -1, -i).
%F A062327 For example, 5 has divisors 1, 1+2i, 2+i and 5.
%K A062327 nonn,nice
%O A062327 1,2
%A A062327 Reiner Martin (reinermartin@hotmail.com), Jul 12 2001
%I A049820
%S A049820 0,0,1,1,3,2,5,4,6,6,9,6,11,10,11,11,15,12,17,14,17,18,21,16,22,
%T A049820 22,23,22,27,22,29,26,29,30,31,27,35,34,35,32,39,34,41,38,39,42,
%U A049820 45,38,46,44,47,46,51,46,51,48,53,54,57,48,59,58,57,57,61,58,65
%N A049820 n-d(n) where d=A000005.
%C A049820 a(n)=number of positive numbers in n-th row of array T given by A049816.
%p A049820 with(numtheory); A049820:=n->n-sigma[0](n);
%Y A049820 Cf. A000005, A062249.
%K A049820 nonn,easy
%O A049820 1,5
%A A049820 Clark Kimberling, ck6@cedar.evansville.edu
%I A054068
%S A054068 1,3,2,5,4,6,8,10,7,9,15,12,14,11,13,20,17,19,16,21,18,26,23,28,25,22,
%T A054068 27,24,33,30,35,32,29,34,31,36,41,38,43,40,45,37,42,39,44,50,55,47,52,
%U A054068 49,54,46,51,48,53,60,65,57,62,59,64,56,61
%N A054068 Permutation of N = set of natural numbers: a(n)+C(k,2), where a=A054065 and k=Floor((1+sqrt(8n-3))/2).
%H A054068 Index entries for sequences that are permutations of the natural numbers
%e A054068 Referring to A054065, just add C(k,2) to the numbers in p(k): e.g., p(1)=(1)->(1); p(2)=(2,1)->(3,2); p(3)=(2,1,3)->(5,4,6).
%Y A054068 Inverse permutation: A054069.
%K A054068 nonn
%O A054068 1,2
%A A054068 Clark Kimberling, ck6@cedar.evansville.edu
%I A054069
%S A054069 1,3,2,5,4,6,9,7,10,8,14,12,15,13,11,19,17,21,18,16,20,26,23,28,25,22,
%T A054069 27,24,33,30,35,32,29,34,31,36,42,38,44,40,37,43,39,45,41,52,48,54,50,
%U A054069 46,53,49,55,51,47,62,58,65,60,56,63,59,66
%N A054069 Inverse of the permutation A054068 of natural numbers.
%H A054069 Index entries for sequences that are permutations of the natural numbers
%K A054069 nonn
%O A054069 1,2
%A A054069 Clark Kimberling, ck6@cedar.evansville.edu
%I A004442
%S A004442 1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14,17,16,19,18,21,20,23,22,25,
%T A004442 24,27,26,29,28,31,30,33,32,35,34,37,36,39,38,41,40,43,42,45,44,47,
%U A004442 46,49,48,51,50,53,52,55,54,57,56,59,58,61,60,63,62,65,64,67,66,69
%N A004442 The natural numbers, pairs reversed: a(n) = n + (-1)^n; also Nimsum n + 1.
%C A004442 A self-inverse permutation of the natural numbers.
%D A004442 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
%D A004442 J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%H A004442 Index entries for sequences related to Nim-sums
%H A004442 Index entries for sequences that are permutations of the natural numbers
%Y A004442 Cf. A003987, A004443, A004444. Equals A014681 - 1.
%K A004442 nonn,easy,nice
%O A004442 0,3
%A A004442 njas
%I A065190
%S A065190 1,3,2,5,4,7,6,9,8,11,10,13,12,15,14,17,16,19,18,21,20,23,22,25,24,27,
%T A065190 26,29,28,31,30,33,32,35,34,37,36,39,38,41,40,43,42,45,44,47,46,49,48,
%U A065190 51,50,53,52,55,54,57,56,59,58,61,60,63,62,65,64,67,66,69,68,71,70,73
%N A065190 Self-inverse permutation of natural numbers: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1).
%H A065190 Index entries for sequences that are permutations of the natural numbers
%F A065190 a(1) = 1, a(n) = n+((-1)^n))
%p A065190 [seq(f(j),j=1..120)]; f := (n) -> `if`((n < 2),n,n+((-1)^n));
%Y A065190 Cf. A004442, A065190 o A014681 = A065168, A014681 o A065190 = A065164.
%K A065190 nonn
%O A065190 1,2
%A A065190 Antti.Karttunen@iki.fi Oct 19 2001
%I A054430
%S A054430 1,3,2,5,4,9,8,7,6,11,10,17,16,15,14,13,12,21,20,19,18,27,26,25,24,23,
%T A054430 22,31,30,29,28,41,40,39,38,37,36,35,34,33,32,45,44,43,42,57,56,55,54,
%U A054430 53,52,51,50,49,48,47,46,63,62,61,60,59,58,71,70,69,68,67,66,65,64,79
%N A054430 Simple self-inverse permutation of natural numbers: List each clump of phi(n) numbers (starting from phi(2) = 1) in reverse order.
%H A054430 Index entries for sequences that are permutations of the natural numbers
%p A054430 ReverseNextPhi_n_elements_permutation(30); with(numtheory,phi); ReverseNextPhi_n_elements_permutation := proc(u) local m,a,n,k,i; a := []; k := 0; for n from 2 to u do m := k + phi(n); for i from 1 to phi(n) do a := [op(a),m]; m := m-1; k := k+1; od; od; RETURN(a); end;
%Y A054430 Maps fractions between A020652/A020653 and A020653/A020652.
%Y A054430 See also A054429.
%K A054430 nonn,easy
%O A054430 1,2
%A A054430 Antti.Karttunen@iki.fi (karttu@megabaud.fi)
%I A054158
%S A054158 1,3,2,5,4,10,9,21,26,48,70,136,209,389,673,1235,2221,4144,7631,
%T A054158 14358,26941,51016,96783,184560,352454,675391,1296503,2494071,4805389,
%U A054158 9273501,17919559,34670835,67156871,130218219,252741267,490988734
%N A054158 Inverse Mobius transform of A001371 (starting at term 0).
%K A054158 nonn
%O A054158 0,2
%A A054158 njas, Apr 29 2000
%I A054160
%S A054160 1,3,2,5,4,10,9,21,26,48,70,136,209,389,673,1235,2221,4144,7631,
%T A054160 14358,26941,51016,96783,184560,352454,675391,1296503,2494071,4805389,
%U A054160 9273501,17919559,34670835,67156871,130218219,252741267,490988734
%N A054160 Mobius transform of A001371 (starting at term 0).
%K A054160 nonn
%O A054160 0,2
%A A054160 njas, Apr 29 2000
%I A054080
%S A054080 1,3,2,5,4,10,10,23,32,62,100,198,336,642,1166,2205,4081,7750,14533,
%T A054080 27658,52388,99960,190558,364938,698874,1342514,2580827,4971652,
%U A054080 9586396,18514020,35790268,69275871,134215781,260305069,505286428
%N A054080 Inverse Mobius transform of A001037 (starting at term 0).
%K A054080 nonn
%O A054080 0,2
%A A054080 njas, Apr 29 2000
%I A001598 M2244 N0891
%S A001598 1,1,3,2,5,5,4,2,9,5,8,5,13,12,8,5,17,8,6,11,14,11,23,7,23,26,11,16,
%T A001598 14,15,31,10,28,16,24,15,37,9,39,16,20,27,20,31,14,43,47,23,32,20,51,17
%N A001598 Number of terms in {b(1)..b(n)} relatively prime to b(n), where b(n) = A001597(n).
%D A001598 H. W. Gould, Problem H-170, Fib. Quart., 8 (1970), 268.
%K A001598 nonn,easy
%O A001598 1,3
%A A001598 njas
%I A059319
%S A059319 1,3,2,5,5,6,3,11,4,15,7,10,9,9,8,21,11,12,15,25,16,21,9,22,7,27,12,15,
%T A059319 17,24,13,43,14,33,19,20,23,45,14,55,17,48,29,35,24,27,19,42,21,21,28,
%U A059319 45,27,36,15,33,14,51,25,40,37,39,30,85,33,42,43,55,34,57,19,44,23,69
%N A059319 Number of 1's in row n of Pascal's rhombus mod 2.
%D A059319 J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 321-236.
%Y A059319 Cf. A059317, A059318, A059320, A007318.
%K A059319 easy,nonn
%O A059319 0,2
%A A059319 njas, Jan 26 2001
%E A059319 More terms from Larry Reeves (larryr@acm.org), Jan 30 2001
%I A019828
%S A019828 3,2,5,5,6,8,1,5,4,4,5,7,1,5,6,6,6,8,7,1,4,0,0,8,9,3,5,7,9,4,7,2,1,
%T A019828 5,7,1,7,9,8,8,5,1,6,0,6,7,5,9,1,2,3,1,0,7,2,1,5,2,2,2,7,9,4,9,4,6,
%U A019828 6,0,1,6,8,0,5,2,8,3,8,4,1,7,9,4,2,8,1,7,6,5,9,6,7,1,3,1,2,7,2,6,9
%N A019828 Decimal expansion of sine of 19 degrees.
%K A019828 nonn,cons
%O A019828 0,1
%A A019828 njas
%I A008623
%S A008623 1,0,1,1,3,2,5,5,9,9,14,15,22,23,32,34,45,48,61,65,81,
%T A008623 87,104,112,133,142,165,177,204,217,247,263,297,315,352,
%U A008623 374,415,439,484,512,561,592,646,680,739,777,840,882
%N A008623 Molien series of 4-dimensional representation of SL(2,7).
%D A008623 C. L. Mallows and N. J. A. Sloane, On invariants of a linear group of order 336, Math. Proc. Camb Phil Soc., 74 (1973), 435-439.
%p A008623 (x^28+x^20+x^18+x^16+x^12+x^10+x^8+1)/(1-x^4)/(1-x^6)/(1-x^8)/(1-x^14);
%K A008623 nonn,easy
%O A008623 0,5
%A A008623 njas
%I A035546
%S A035546 0,0,0,0,0,1,0,1,1,3,2,5,5,9,9,16,17,26,30,43,50,68,81,108,128,165,200,
%T A035546 253,302,378,456,560,672,821,984,1190,1421,1708,2036,2432,2886,3433,4064,
%U A035546 4807,5673,6690,7866,9241,10839,12689,14839,17324,20201,23518,27355
%N A035546 Partitions of n with equal nonzero number of parts congruent to each of 0 and 1 (mod 4)
%K A035546 nonn,part
%O A035546 0,10
%A A035546 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A026098
%S A026098 1,3,2,5,6,4,7,10,9,8,11,14,15,12,16,13,22,21,20,18,32,17,26,33,28,25,64,19,34,39,
%T A026098 44,35,30,27,128,23,38,51,52,55,42,40,36,256,29,46,57,68,65,66,49,45,48,512,31,58,
%U A026098 69,76,85,78,77,56,50,54,1024,37,62,87,92,95,102,91,88,63,60,72,2048,41,74,93,116,115
%N A026098 Triangular array T by rows: T(1,1)=1, T(2,1)=3, T(2,2)=2; for n >= 3, T(n,1)=p(n), and for k=2,3,...,n, T(n,k) = m*p(n+1-k), where m is the least positive integer such that q*p(n+1-k) is not any T(i,j) for 1<=i<=n-1 nor any T(n,j) for j<=k-1.
%K A026098 nonn
%O A026098 1,2
%A A026098 Clark Kimberling (ck6@cedar.evansville.edu)
%I A057028
%S A057028 1,3,2,5,6,4,9,8,10,7,13,14,12,15,11,19,18,20,17,21,16,25,26,24,27,23,
%T A057028 28,22,33,32,34,31,35,30,36,29,41,42,40,43,39,44,38,45,37,51,50,52,49,
%U A057028 53,48,54,47,55,46,61,62,60,63,59,64,58,65,57,66,56
%N A057028 Triangle T by rows: row n consists of the numbers C(n,2)+1 to C(n+1,2); numbers in odd-numbered places form a decreasing sequence, and the others an increasing sequence.
%H A057028 Index entries for sequences that are permutations of the natural numbers
%e A057028 Rows: 1; 3,2; 5,6,4; 9,8,10,7; ...
%Y A057028 Reflection of the array in A057027 about its central column; a permutation of the natural numbers.
%Y A057028 Inverse permutation: A064789.
%K A057028 nonn,tabl
%O A057028 1,2
%A A057028 Clark Kimberling, ck6@evansville.edu, Jul 28, 2000
%I A055922
%S A055922 1,1,1,3,2,5,6,9,9,16,20,25,32,40,54,69,84,101,136,156,202,244,306,357,
%T A055922 448,527,652,773,944,1103,1346,1574,1885,2228,2640,3106,3684,4302,5052,
%U A055922 5931,6924,8079,9416,10958,12718,14824,17078,19820,22860,26433
%N A055922 Partitions of n in which each part occurs an odd number (or 0) times.
%F A055922 EULER transform of b where b has g.f. SUM {k>0} c(k)*x^k/(1-x^k) where c is inverse EULER transform of characteristic function of odd numbers.
%Y A055922 Cf. A000041, A007690, A055923.
%K A055922 nonn,part
%O A055922 0,4
%A A055922 Christian G. Bower (bowerc@usa.net), Jun 23 2000
%I A050061
%S A050061 1,3,2,5,6,11,13,16,17,33,46,57,63,68,70,73,74,147,217,285,348,405,451,
%T A050061 484,501,517,530,541,547,552,554,557,558,1115,1669,2221,2768,3309,3839,
%U A050061 4356,4857,5341,5792,6197,6545,6830,7047,7194
%N A050061 a(n)=a(n-1)+a(m), where m=2^(p+1)+2-n, and 2^p= 4.
%K A050061 nonn
%O A050061 1,2
%A A050061 Clark Kimberling, ck6@cedar.evansville.edu
%I A058638
%S A058638 1,0,3,2,5,6,12,12,22,22,39,40,63,68,106,112,164,182,257,282,390,
%T A058638 432,584,652,859
%N A058638 McKay-Thompson series of class 34A for Monster.
%D A058638 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058638 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058638 nonn
%O A058638 -1,3
%A A058638 njas, Nov 27, 2000
%I A047074
%S A047074 1,1,1,3,2,5,6,14,20,45,70,154,252,546,924,1980,3432,7293,12870,27170,
%T A047074 48620,102102,184756,386308,705432,1469650,2704156,5616324,10400600,
%U A047074 21544100,40116600,82907640,155117520,319929885,601080390,1237518450
%N A047074 SUM{T(i,n-i): i=0,1,...,[ n/2 ]}, array T as in A047072.
%K A047074 nonn
%O A047074 0,4
%A A047074 Clark Kimberling, ck6@cedar.evansville.edu
%I A021311
%S A021311 0,0,3,2,5,7,3,2,8,9,9,0,2,2,8,0,1,3,0,2,9,3,1,5,9,6,0,9,1,2,0,5,2,
%T A021311 1,1,7,2,6,3,8,4,3,6,4,8,2,0,8,4,6,9,0,5,5,3,7,4,5,9,2,8,3,3,8,7,6,
%U A021311 2,2,1,4,9,8,3,7,1,3,3,5,5,0,4,8,8,5,9,9,3,4,8,5,3,4,2,0,1,9,5,4,3
%N A021311 Decimal expansion of 1/307.
%K A021311 nonn,cons
%O A021311 0,3
%A A021311 njas
%I A006369 M2245
%S A006369 0,1,3,2,5,7,4,9,11,6,13,15,8,17,19,10,21,23,12,25,27,14,
%T A006369 29,31,16,33,35,18,37,39,20,41,43,22,45,47,24,49,51,26,
%U A006369 53,55,28,57,59,30,61,63,32,65,67,34,69,71,36,73,75,38
%N A006369 Nearest integer to 4n/3 unless that is an integer, when 2n/3.
%D A006369 R. K. Guy, Unsolved Problems in Number Theory, E17.
%H A006369 J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
%H A006369 Index entries for sequences that are permutations of the natural numbers
%F A006369 Expansion of (1+3x+2x^2+3x^3+x^4)/(1-x^3)^2.
%p A006369 A006369:=proc(n) if n mod 3 = 0 then 2*n/3 else round(4*n/3); fi; end;
%Y A006369 Inverse mapping to A006368.
%K A006369 nonn,nice,easy
%O A006369 0,3
%A A006369 njas, jhc
%I A013655
%S A013655 3,2,5,7,12,19,31,50,81,131,212,343,555,898,1453,2351,3804,6155,9959,
%T A013655 16114,26073,42187,68260,110447,178707,289154,467861,757015,1224876,
%U A013655 1981891,3206767,5188658,8395425,13584083,21979508,35563591,57543099
%N A013655 F(n)+L(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.
%F A013655 a(n)=a(n-1)+a(n-2).
%K A013655 nonn,easy
%O A013655 0,1
%A A013655 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A013655 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A016649
%S A016649 3,2,5,8,0,9,6,5,3,8,0,2,1,4,8,2,0,4,5,4,7,0,7,1,9,5,6,3,0,2,3,4,9,
%T A016649 5,1,7,2,8,8,0,7,6,8,0,7,9,1,2,0,4,6,2,3,7,0,5,3,9,7,2,5,5,2,0,1,5,
%U A016649 6,8,5,8,2,8,9,2,9,4,1,0,4,8,9,5,0,0,5,4,3,7,9,9,0,4,3,6,9,0,8,1,7
%N A016649 Decimal expansion of ln(26).
%D A016649 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
%K A016649 nonn,cons
%O A016649 1,1
%A A016649 njas
%I A019594
%S A019594 1,3,2,5,8,5,9,5,10,15,9,15,21,13,20
%N A019594 Conway's "para-budding" sequence.
%D A019594 J. H. Conway, personal communication.
%H A019594 N. J. A. Sloane, Classic Sequences
%K A019594 nonn
%O A019594 0,2
%A A019594 njas
%I A029619
%S A029619 1,3,2,5,8,7,11,15,9,14,26,24,17,40,50,35,13,20,57,90,85,48,23,77,147,
%T A029619 175,133,63,100,224,322,308,196,80,19,29,126,324,546,630,504,276,99,21,
%U A029619 32,155,450,870,1176,1134,780,375,120,187,605,1320,2046,2310,1914,1155
%N A029619 Distinct numbers in (3,2)-Pascal triangle A029618.
%K A029619 nonn
%O A029619 0,2
%A A029619 Mohammad K. Azarian, ma3@cedar.evansville.edu
%E A029619 More terms from James A. Sellers (sellersj@math.psu.edu), Dec 08 1999
%I A049922
%S A049922 1,3,2,5,8,18,34,69,135,274,546,1093,2183,4363,8716,17416,34797,69662,
%T A049922 139322,278645,557287,1114571,2229132,4458248,8916461,17832856,
%U A049922 35665573,71330874,142661201,285321312,570640444,1141276535,2282544370
%N A049922 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=n-1-2^p, and 2^p= 4.
%K A050063 nonn
%O A050063 1,2
%A A050063 Clark Kimberling, ck6@cedar.evansville.edu
%I A062941
%S A062941 3,2,5,12,25,53,116,249,535,1155,2487,5358,11545,24871,53584,115444,
%T A062941 248715,535841,1154435,2487154,5358411,11544347,24871542,53584111,
%U A062941 115443470,248715414,535841116,1154434691,2487154143,5358411166
%N A062941 Number of n-digit cubes (0 is included as a single digit number).
%C A062941 Sum of first 3n terms = 10^n.
%e A062941 a(3) = 5 as there are 5 three-digit cubes: 125, 216, 343, 512 and 729.
%Y A062941 A062940.
%K A062941 base,nonn
%O A062941 0,1
%A A062941 Amarnath Murthy (amarnath_murthy@yahoo.com), Jul 07 2001
%E A062941 Corrected and extended by Dean Hickerson (dean@math.ucdavis.edu), Jul 10, 2001
%I A057674
%S A057674 3,2,5,13,29,61,109,251,509,1021,2029,4093,8179,16381,32749,65519,
%T A057674 131059,262139,524269,1048573,2097133,4194301,8388571,16777213,
%U A057674 33554371,67108859,134217649,268435367,536870909,1073741783,2147483629
%V A057674 -3,2,5,13,29,61,109,251,509,1021,2029,4093,8179,16381,32749,65519,131059,262139,
%W A057674 524269,1048573,2097133,4194301,8388571,16777213,33554371,67108859,134217649,
%X A057674 268435367,536870909,1073741783,2147483629,4294967291,8589934513,17179869143
%N A057674 Primes -p+2^n with smallest p prime, arising in A057674.
%e A057674 n=1, 2^1=2. If 2,3,5 are subtracted from 2, then 0,-1 and -3 arise, of which -3 is a prime so a(1)=-3. n=11, 2048-p=q. At first p=29 gives the prime q=2029.
%Y A057674 Cf. A056206, A056208, A056662, A057674.
%K A057674 sign,done
%O A057674 0,1
%A A057674 Labos E. (labos@ana1.sote.hu), Oct 19 2000
%I A005265 M2246
%S A005265 3,2,5,29,11,7,13,37,32222189,131,136013303998782209,31,197,19,157,
%T A005265 17,8609,1831129,35977,508326079288931,487,10253,1390043,18122659735201507243,25319167,9512386441,85577,1031,
%U A005265 3650460767,107,41,811,15787,89,68168743,4583,239,1283,443,902404933,64775657,2753,23,149287,149749,7895159,79,43,1409,184274081,47,569,63843643
%N A005265 From Euclid's proof.
%D A005265 R. K. Guy and R. Nowakowski, ``Discovering primes with Euclid,'' Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
%D A005265 S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.
%F A005265 a(1)=3, b(n)=Product_{k=1..n} a(k), a(n+1)=least prime factor of b(n)-1.
%Y A005265 Cf. A000945, A000946, A005266.
%K A005265 nonn,nice
%O A005265 0,1
%A A005265 njas
%I A005266 M2247
%S A005266 3,2,5,29,79,68729,3739,6221191,157170297801581,70724343608203457341903,
%T A005266 46316297682014731387158877659877,78592684042614093322289223662773,181891012640244955605725966274974474087,
%U A005266 547275580337664165337990140111772164867508038795347198579326533639132704344301831464707648235639448747816483406685904347568344407941
%N A005266 From Euclid's proof.
%D A005266 R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
%D A005266 S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.
%F A005266 a(1)=3, b(n)=Product_{k=1..n} a(k), a(n+1)=greatest prime factor of b(n)-1.
%Y A005266 Cf. A000945, A000946, A005265.
%K A005266 nonn,nice,huge
%O A005266 0,1
%A A005266 njas
%E A005266 a(14) from Joe Crump (joecr@microsoft.com), Jul 26, 2000
%I A005267 M2248
%S A005267 3,2,5,29,869,756029,571580604869,326704387862983487112029,
%T A005267 106735757048926752040856495274871386126283608869
%N A005267 a(n+1) = a(n)^2 + a(n) - 1 (n>0).
%D A005267 R. K. Guy and R. Nowakowski, ``Discovering primes with Euclid,'' Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
%F A005267 a(n) = -1 + a(0)a(1)...a(n-1).
%F A005267 a(n) = -1 + product_{i= 0.
%C A061187 pLo(m+1,x) appears as numerator polynomial of the g.f. for column m >= 0 of the triangle A060924 (even part of bisection of Lucas triangle).
%F A061187 a(n,m)= sum(3*(-9/2)^j*binomial(n+1,2*j+1)*sum((-3/2)^(k-m)*binomial(n-2*j,k) *binomial(2*j+1,m-k-j),k=max(0,m-3*j-1)..n-2*j),j=0..floor(n/2)), 0<= m <= n+1+floor(n/2); else 0.
%e A061187 {3, -2}; {6, 2, -4}; {9, 39, -57, 30, -8}; ...; pLo(2, x)= 6+2*x-4*x^2= 2*(1+x)*(3-2*x).
%Y A061187 A061186 (companion staircase).
%K A061187 sign,done,easy,tabf
%O A061187 0,1
%A A061187 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Apr 20 2001
%I A021758
%S A021758 0,0,1,3,2,6,2,5,9,9,4,6,9,4,9,6,0,2,1,2,2,0,1,5,9,1,5,1,1,9,3,6,3,
%T A021758 3,9,5,2,2,5,4,6,4,1,9,0,9,8,1,4,3,2,3,6,0,7,4,2,7,0,5,5,7,0,2,9,1,
%U A021758 7,7,7,1,8,8,3,2,8,9,1,2,4,6,6,8,4,3,5,0,1,3,2,6,2,5,9,9,4,6,9,4,9
%N A021758 Decimal expansion of 1/754.
%K A021758 nonn,cons
%O A021758 0,4
%A A021758 njas
%I A040008
%S A040008 3,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,
%T A040008 2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,
%U A040008 6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6
%N A040008 Continued fraction for sqrt(12).
%H A040008 Index entries for continued fractions for constants
%p A040008 Digits:=100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
%K A040008 nonn,cofr,easy
%O A040008 0,1
%A A040008 njas
%I A065228
%S A065228 1,1,3,2,6,2,8,15,7,16,1,12,24,1,15,30,46,14,32,51,7,28,50,73,16,41,67,
%T A065228 94,22,51,81,112,23,56,90,125,17,54,92,131,2,43,85,128,172,21,67,114,
%U A065228 162,211,36,87,139,192,246,45,101,158,216,275,46,107,169,232,296,37
%N A065228 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the square numbers. The first elements of the rows form a(n).
%Y A065228 Cf. A064766, A064865, A065221-A065227, A065229-A065234.
%K A065228 easy,nonn
%O A065228 0,3
%A A065228 Floor van Lamoen (f.v.lamoen@wxs.nl), Oct 22 2001
%I A058971
%S A058971 3,2,6,3,3,4,10,87,6,6,9,7,6,6,87,9,6,10,7,8,9,12,9,15,12,10,16,15,9,
%T A058971 16,12,12,15,12,87,19,15,14,19,21,12,22,14,13,18,24,34,19,12,18,0,27,
%U A058971 15,18,15,20,24,30,14,31,24,18,51,21,18,34,21,24,18,36,24,37,30,21,37
%N A058971 For a rational number p/q let f(p/q) = sum of divisors of p+q divided by number of divisors of p+q; a(n) is obtained by iterating f, starting at n/1, until an integer is reached, or if no integer is ever reached then a(n) = 0.
%e A058971 1 -> (1+2)/2 = 3/2 -> (1+5)/2 = 3, so a(1) = 3. 51 -> 49/3 -> 49/3 -> ..., so a(51) = 0.
%p A058971 with(numtheory); f:=proc(n) if whattype(n) = integer then sigma(n+1)/sigma[0](n+1) else sigma(numer(n)+denom(n))/sigma[0](numer(n)+denom(n)); fi; end;
%Y A058971 Cf. A058972, A058977.
%K A058971 nonn,easy,nice
%O A058971 1,1
%A A058971 njas, Jan 14 2001
%E A058971 More terms from Matthew M. Conroy (doctormatt@earthlink.net), Apr 18 2001, who remarks that a(51) = a(655) = a(1039) = 0 are all the zeros of a(n) for n<10^5.
%I A011209
%S A011209 1,1,1,3,2,6,3,5,7,6,8,4,4,8,0,3,3,9,4,1,4,5,9,8,7,2,8,1,9,4,0,2,3,
%T A011209 0,5,5,6,7,5,7,9,2,2,2,1,3,8,6,4,3,1,1,9,0,7,3,4,6,2,1,0,7,9,0,3,1,
%U A011209 0,5,9,1,6,8,8,1,1,6,1,3,7,0,0,9,3,7,5,4,6,6,4,2,4,2,2,4,7,0,2,9,3
%N A011209 Decimal expansion of 15th root of 5.
%K A011209 nonn,cons
%O A011209 1,4
%A A011209 njas
%I A038572
%S A038572 0,1,1,3,2,6,3,7,4,12,5,13,6,14,7,15,8,24,9,25,10,26,11,27,12,28,13,29,
%T A038572 14,30,15,31,16,48,17,49,18,50,19,51,20,52,21,53,22,54,23,55,24,56,25,
%U A038572 57,26,58,27,59,28,60,29,61,30,62,31,63,32,96,33,97,34,98,35,99,36,100
%N A038572 n rotated one binary place to the right.
%F A038572 a(n) = A053645(n) * A000035(n) + A004526(n) = msb(n) * lsb(n) + floor(n/2).
%Y A038572 Cf. A006257.
%K A038572 easy,nonn,nice
%O A038572 0,4
%A A038572 Marc Le Brun (mlb@well.com)
%I A064455
%S A064455 1,3,2,6,3,9,4,12,5,15,6,18,7,21,8,24,9,27,10,30,11,33,12,36,13,39,14,
%T A064455 42,15,45,16,48,17,51,18,54,19,57,20,60,21,63,22,66,23,69,24,72,25,75,
%U A064455 26,78,27,81,28,84,29,87,30,90,31,93,32,96,33,99,34,102,35,105,36,108
%N A064455 a(2n) = 3n, a(2n-1) = n.
%e A064455 a(13) = a(2*7 - 1) = 7. a(14) = a(2*7) = 21.
%t A064455 Table[ If[ EvenQ[n], 3n/2, (n + 1)/2], {n, 1, 70} ]
%o A064455 (ARIBAS): maxarg := 75; for n := 1 to maxarg do if n mod 2 = 1 then write((n+1) div 2, " ") else write((n div 2)*3," "); end; end;.
%Y A064455 Interleaving of A000027 and A008585 (without first term). Cf. A064433.
%K A064455 nonn,easy
%O A064455 1,2
%A A064455 njas, Oct 02 2001
%E A064455 More terms from Klaus Brockhaus (klaus-brockhaus@t-online.de) and Robert G. Wilson v (rgwv@kspaint.com), Oct 03 2001
%I A065021
%S A065021 3,2,6,4,1,2,1,9,2,1,2,3,2,3,5,1,2,1,1,6,1,2,5,79,6,4,5,1,1,1,1,12,1,1,
%T A065021 2,5,1,659,2,12,1,862,1,12,1,4,2,2,8
%N A065021 Continued fraction expansion of the constant Product{k=1..inf} (1-1/2^k)^(-1) (A065446).
%t A065021 ContinuedFraction[ N[ Product[ 1/(1 - 1/2^k), {k, 1, Infinity} ], 500 ], 49]
%Y A065021 Cf. A065446.
%K A065021 cofr,nonn
%O A065021 1,1
%A A065021 Robert G. Wilson v (rgwv@kspaint.com), Nov 19 2001
%I A048652
%S A048652 0,3,2,6,4,1,2,1,9,2,1,2,3,2,3,5,1,2,1,1,6,1,2,5,79,6,4,5,1,1,1,
%T A048652 1,101,46,1,43,1,2,7,23,2,1,18,1,1,1,8,4,7,9,1,1,1,1,3,3,4,76,1,
%U A048652 31,1,3,72,1,1,4,4,6,4,1,1,7,8,1,36,18,2,31,1,2,7,1,12,3,1,1,15
%N A048652 Continued fraction for Product_{k >= 1} (1-1/2^k).
%H A048652 S. Finch, Digital Search Tree Constants
%H A048652 Index entries for continued fractions for constants
%e A048652 0.2887880950866024212788997219294585937270...
%Y A048652 Cf. A005329, A048651.
%K A048652 nonn,cofr
%O A048652 0,2
%A A048652 njas
%I A057050
%S A057050 1,3,2,6,4,1,7,3,11,6,16,10,3,15,7,21,12,2,18,7,25,13,33,20,6,28,13,37,
%T A057050 21,4,30,12,40,21,1,31,10,42,20,54,31,7,43,18,56,30,3,43,15,57,28,72,
%U A057050 42,11,57,25,73,40,6,56,21,73,37,91,54,16,72
%N A057050 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6;...; each k is an R(i(k),j(k)), and A057050(n)=j(n^2).
%K A057050 nonn
%O A057050 1,2
%A A057050 Clark Kimberling, ck6@cedar.evansville.edu, Jul 30 2000
%I A033940
%S A033940 1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,
%T A033940 1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,
%U A033940 5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6,4,5,1,3,2,6
%N A033940 10^n mod 7.
%K A033940 nonn
%O A033940 0,2
%A A033940 Jeff Burch (gburch@erols.com)
%I A059399
%S A059399 1,3,2,6,4,5,10,7,9,8,15,11,14,12,13,21,16,20,17,19,18,28,22,27,23,26,
%T A059399 24,25,36,29,35,30,34,31,33,32,45,37,44,38,43,39,42,40,41,55,46,54,47,
%U A059399 53,48,52,49,51,50,66,56,65,57,64,58,63,59,62,60,61
%N A059399 Triangular hopscotch.
%F A059399 Write down triangle with rows 1; 2,3; 4,5,6; 7,8,9,10; ...; read rows alternately from right and left ends.
%K A059399 nonn,easy
%O A059399 1,2
%A A059399 Robert G. Wilson v (rgwv@kspaint.com), Jan 29 2001
%I A064789
%S A064789 1,3,2,6,4,5,10,8,7,9,15,13,11,12,14
%N A064789 Inverse permutation to A057028.
%H A064789 Index entries for sequences that are permutations of the natural numbers
%K A064789 easy,nonn,more
%O A064789 1,2
%A A064789 njas, Oct 20 2001
%I A054089
%S A054089 1,3,2,6,4,8,5,11,7,14,9,16,10,19,12,21,13,24,15,27,17,29,18,32,20,35,
%T A054089 22,37,23,40,25,42,26,45,28,48,30,50,31,53,33,55,34,58,36,61,38,63,39,
%U A054089 66,41,69,43,71,44,74,46,76,47,79,49,82,51
%N A054089 For k >= 1, let p(k)=least h in N not already an a(i), q(k)=p(k)+k, a(2k)=q(k), a(2k+1)=p(k).
%Y A054089 Odd-indexed terms, Cf. A026351(n)=[n*tau]+1.
%Y A054089 Even-indexed terms, Cf. A004957(n)=[n*tau]+n+1.
%K A054089 nonn
%O A054089 1,2
%A A054089 Clark Kimberling, ck6@cedar.evansville.edu
%I A006368 M2249
%S A006368 1,3,2,6,4,9,5,12,7,15,8,18,10,21,11,24,13,27,14,30,16,33,17,36,19,39,
%T A006368 20,42,22,45,23,48,25,51,26,54,28,57,29,60,31,63,32,66,34,69,35,72,37,
%U A006368 75,38,78,40,81,41,84,43,87,44,90,46,93,47,96,49,99,50,102,52,105,53
%N A006368 If n even then 3n/2 otherwise nearest integer to 3n/4.
%D A006368 R. K. Guy, Unsolved Problems in Number Theory, E17.
%F A006368 Expansion of (1+3x+x^2+3x^3+x^4)/(1-x^2)(1-x^4).
%o A006368 (PARI.2.0.17) j(e)=if(Mod(n,2)==0,(3*n/2),round(3*n/4)); vector(200,n,j(e))
%Y A006368 Inverse mapping to A006369.
%K A006368 nonn,nice,easy
%O A006368 0,2
%A A006368 njas, jhc
%E A006368 More terms from Jason Earls (jcearls@kskc.net), Jul 12 2001
%I A049777
%S A049777 1,3,2,6,5,3,10,9,7,4,15,14,12,9,5,21,20,18,15,11,6,28,27,25,22,18,13,
%T A049777 7,36,35,33,30,26,21,15,8,45,44,42,39,35,30,24,17,9,55,54,52,49,45,40,
%U A049777 34,27,19,10,66,65,63,60,56,51,45,38,30,21
%N A049777 Triangular array T by rows: T(m,n) = n + n+1 + ... + m = (m+n)(m-n+1)/2.
%e A049777 Rows: {1}; {3,2}; {6,5,3}; ...
%Y A049777 Row sums = A000330.
%K A049777 nonn,tabl
%O A049777 1,2
%A A049777 Clark Kimberling, ck6@cedar.evansville.edu
%I A058401
%S A058401 1,3,2,6,5,3,13,12,9,5,22,21,18,13,7,42,41,37,30,21,11,66,65,61,53,42,
%T A058401 29,15,112,111,106,96,81,63,43,22,172,171,166,154,136,113,87,59,30,270,
%U A058401 269,263,249,226,196,161,123,83,42,397,396,390,374,347,310,266,217,165
%N A058401 Triangle of partial row sums of partition triangle A026820.
%C A058401 m=1 column: A058397(n).
%F A058401 a(n,m)= sum(A026820(n,k),k=m..n).
%e A058401 1; 3,2; 6,5,3; 13,12,9,5; ...
%Y A058401 Cf. A026820, A058397.
%K A058401 nonn,tabl
%O A058401 1,2
%A A058401 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Dec 11 2000
%I A038722
%S A038722 1,3,2,6,5,4,10,9,8,7,15,14,13,12,11,21,20,19,18,17,16,28,27,26,25,
%T A038722 24,23,22,36,35,34,33,32,31,30,29,45,44,43,42,41,40,39,38,37,55,54,
%U A038722 53,52,51,50,49,58,47,46,66,65,64,63,62,61,60,59,58,57,56,78,77,76
%N A038722 Take sequence of natural numbers (A000027), and reverse successive subsequences of lengths 1,2,3,4,...
%D A038722 Suggested by correspondence with Michael Somos (somos@grail.cba.csuohio.edu).
%D A038722 R. Honsberger, "Ingenuity in Mathematics", Table 10.4 on page 87.
%H A038722 Index entries for sequences that are permutations of the natural numbers
%F A038722 a(n) =[sqrt(2n-1)-1/2]*[sqrt(2n-1)+3/2]-n+2 =A061579(n-1)+1. Seen as a square table by anti-diagonals, T(n,k)=k+(n+k-1)*(n+k-2)/2, i.e. the transpose of A000027 as a square table.
%Y A038722 A self-inverse permutation of the natural numbers.
%Y A038722 Cf. A000027, A020703.
%K A038722 nonn,tabl
%O A038722 1,2
%A A038722 njas, May 02 2000
%I A046877
%S A046877 3,2,6,5,8,11,13,19,24,32,43,56,75,99,131,174,230,305,404,535,709,
%T A046877 939,1244,1648,2183,2892,3831,5075,6723,8906,11798,15629,20704,
%U A046877 27427,36333,48131,63760,84464,111891,148224,196355,260115,344579
%N A046877 a(n)=a(n-2)+a(n-3).
%F A046877 G.f.: (3+2*x+3*x^2)/(1-x^2-x^3).
%Y A046877 Cf. A001608.
%K A046877 nonn
%O A046877 0,1
%A A046877 njas
%I A064684
%S A064684 0,3,2,6,6,5,3,4,4,7,1,4,7,25,6,24,7,3,7,11,25,8,4,23,7,6,3,24,8,11,5,
%T A064684 20,7,9,3,22,25,2,6,11,5,24,1,9,10,20,3,20,26,7,8,19,11,21,26,15,2,8,5,
%U A064684 10,20,13,23,12,26,9,7,9,20,13,3,20,26,24,7,3,8,18,12,13,20,5,24,15,12
%N A064684 Number of primes in orbit of 2n+1 in the 3x+1 problem.
%H A064684 Index entries for sequences related to 3x+1 (or Collatz) problem
%e A064684 orbit(3) = 3->10->5->16->8->4->2->1. This contains 3 primes, 3, 5 and 2.
%o A064684 (ARIBAS): function orbit(n: integer): array; var stk: stack; begin stack_push(stk,n); while n <> 1 do if n mod 2 = 0 then n := n div 2; else n := 3*n + 1; end; stack_push(stk,n); end; return stack2array(stk); end; function primesfilter(ar: array): array; var j,k: integer; stk: stack; begin for j := 0 to length(ar) - 1 do k := prime32test(ar[j]); if k = 1 then stack_push(stk,ar[j]); end; end; return stack2array(stk); end; function a064684(maxarg: integer); var n: integer; begin for n := 1 to maxarg by 2 do write(length(primesfilter(orbit(n)))," "); end; end; a064684(190).
%K A064684 nonn,easy
%O A064684 0,2
%A A064684 Jon Perry (perry@globalnet.co.uk), Oct 10 2001
%E A064684 More terms from Klaus Brockhaus (klaus-brockhaus@t-online.de), Oct 13 2001
%I A060408
%S A060408 1,3,2,6,6,6,4,2,10,14,20,24,24,16,8,15,26,48,80,120,144,144,96,48,
%T A060408 21,44,99,212,420,720,1080,1296,1296,864,432
%N A060408 Triangle in which n-th row gives numbers of super edge-magic (n,k) graphs, for n >= 2, k >= 1.
%C A060408 Rows have irregular lengths.
%D A060408 R. M. Figueroa-Centeno et al., The place of super edge-magic labelings among other classes of labelings, Discrete Math., 231 (2001), 153-168.
%e A060408 1; 3,2; 6,6,6,4,2; ...
%K A060408 nonn,tabf
%O A060408 2,2
%A A060408 njas, Apr 06 2001
%I A023360
%S A023360 1,0,1,1,1,3,2,6,6,10,16,20,35,46,72,105,152,232,332,501,732,1081,1604,
%T A023360 2352,3493,5136,7595,11212,16534,24442,36039,53243,78573,115989,171264,
%U A023360 252754,373214,550863,813251,1200554,1772207,2616338,3862121,5701553
%N A023360 Compositions into unordered sums of primes.
%F A023360 a(n) = sum[a(n-p)] over primes p<=n with a(0)=1 - Henry Bottomley (se16@btinternet.com), Dec 15 2000
%K A023360 nonn
%O A023360 0,6
%A A023360 dww
%I A062200
%S A062200 1,1,1,3,2,6,6,11,16,22,37,49,80,113,172,257,377,573,839,1266,1874,
%T A062200 2798,4175,6204,9274,13785,20577,30640,45665,68072,101393,151169,
%U A062200 225193,335659,500162,745342,1110790,1655187,2466760,3675822
%N A062200 Compositions of n such that two adjacent parts are not equal modulo 2.
%C A062200 Also (0,1)-strings such that all maximal blocks of 1's have even length and all maximal blocks of 0's have odd length.
%D A062200 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(Problems 2.4.3, 2.4.13).
%F A062200 a(n)= Sum_{j=0..n+1} binomial(n-j+1,3*j-n+1). a(n) = 2*a(n-2)+a(n-3)-a(n-4).
%F A062200 G.f.: -(x^2-x-1)/(x^4-x^3-2*x^2+1). More generally, g.f. for the number of compositions of n such that two adjacent parts are not equal modulo p is 1/(1-Sum_{i=1..p} x^i/(1+x^i-x^p)).
%Y A062200 Cf. A003242, A062201-A062203.
%K A062200 nonn
%O A062200 0,4
%A A062200 Vladeta Jovovic (vladeta@Eunet.yu), Jun 13 2001
%I A014686
%S A014686 3,2,6,6,12,12,18,18,24,28,32,36,42,42,48,52,60,60,68,70,74,78,84,88,
%T A014686 98,100,104,106,110,112,128,130,138,138,150,150,158,162,168,172,180,
%U A014686 180,192,192,198,198,212,222,228,228,234,238,242,250,258,262,270,270
%N A014686 In sequence of prime numbers add 1 to 1st prime, 3rd prime, fifth prime, ... then subtract 1 from 2nd prime, fourth prime, sixth prime and so on.
%K A014686 nonn,easy
%O A014686 1,1
%A A014686 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A014686 More terms from Andrew J. Gacek (andrew@dgi.net)
%I A053090
%S A053090 1,0,3,2,6,6,12,12,21,22,33,36,50,54,72,78,99,108,133,144,174,188,222,
%T A053090 240,279,300,345,370,420,450,506,540,603,642,711,756,832,882,966,1022,
%U A053090 1113,1176,1275,1344,1452,1528,1644,1728,1853,1944,2079,2178,2322,2430
%N A053090 F^3-convex polyominoes on honeycomb lattice with given semi-perimeter.
%D A053090 Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux decales oscillants . These de Doctorat. Laboratoire de Recherche en Informatique, Universite Paris-Sud XI, France.
%D A053090 Alain Denise, Christoph Durr and Fouad Ibn-Majdoub-Hassani. Enumeration et generation aleatoire de polyominos convexes en reseau hexagonal (French) [enumeration and random generation of convex polyominoes in the honeycomb lattice] . In Proceedings of 9th Conference on Formal Power Series and Algebraic Combinatorics, pages 222-234, 1997.
%F A053090 G.f.: x^3*[(1+x^3)/((1-x^2)^3*(1-x^3))]
%K A053090 nonn
%O A053090 3,3
%A A053090 Fouad Ibn Majdoub Hassani (Fouad.Hassani@genset.fr), Feb 28 2000
%E A053090 More terms from James A. Sellers (sellersj@math.psu.edu), Mar 01 2000
%I A023897
%S A023897 1,3,2,6,7,4,3,9,2,8,5,6,7,4,7,10,5,12,4,9,10,3,4,14,10,8,6,13,9,8,5,15,7,
%T A023897 2,6,8,4,5,12,6,7,10,10,11,14,12,9,4,3,4,12,9,4,4,7,5,7,10,3,5,4,13,14,12,
%U A023897 10,9,10,8,7,4,8,6,18,9,3,8,13,8,15,15,8,3,14,9,10,8,8,10,5,7,8,11,6,11,13,6
%N A023897 sigma_1(n) / phi(n) for balanced numbers.
%C A023897 sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).
%t A023897 Select[ Array[ DivisorSigma[ 1,# ]/EulerPhi[ # ]&,20000 ], IntegerQ ]
%Y A023897 Cf. A000010, A000203, A020492.
%K A023897 nonn
%O A023897 1,2
%A A023897 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A063946
%S A063946 0,1,3,2,6,7,4,5,12,13,14,15,8,9,10,11,24,25,26,27,28,29,30,31,16,17,
%T A063946 18,19,20,21,22,23,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,32,
%U A063946 33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,96,97,98,99,100,101,102
%N A063946 Write n in binary and complement second bit (from the left), with a(0)=0 and a(1)=1.
%F A063946 If 2*2^k<=n<3*2^k then a(n)=n+2^k; if 3*2^k<=n<4*2^k then a(n)=n-2^k.
%e A063946 a(11)=15 since 11 is written in binary as 1011 which changes to 1111 i.e. 15; a(12)=8 since 12 is written as 1100 which changes to 1000 i.e. 8.
%Y A063946 Cf. A004442, A053645, A054429.
%K A063946 easy,nonn
%O A063946 0,3
%A A063946 Henry Bottomley (se16@btinternet.com), Sep 03 2001
%I A003188 M2250
%S A003188 0,1,3,2,6,7,5,4,12,13,15,14,10,11,9,8,24,25,27,26,30,31,29,28,20,21,
%T A003188 23,22,18,19,17,16,48,49,51,50,54,55,53,52,60,61,63,62,58,59,57,56,40,
%U A003188 41,43,42,46,47,45,44,36,37,39,38,34,35,33,32,96,97,99,98,102,103,101
%N A003188 Decimal equivalent of Gray code for n.
%C A003188 Inverse of sequence A006068 considered as a permutation of the non-negative integers, i.e. A006068(A003188(n)) = n = A003188(A006068(n)). - Howard A. Landman (howard@polyamory.org), Sep 25 2001
%D A003188 M. Gardner, Mathematical Games, Sci. Amer. Vol. 227 (No. 2, Feb. 1972), p. 107.
%D A003188 M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 15.
%H A003188 Index entries for sequences that are permutations of the natural numbers
%F A003188 a(n) =2*a([n/2])+A021913(n-1) - Henry Bottomley (se16@btinternet.com), Apr 05 2001
%p A003188 with(combinat); graycode(6); # to produce first 64 terms
%p A003188 printf(cat(` %.6d`$64), op(map(convert, graycode(6), binary))); lprint(); # to produce binary strings
%Y A003188 A003188[ 2*A003714[ n ] ] = 3*A003714[ n ] for all n - Antti.Karttunen@iki.fi (karttu@megabaud.fi), 26.April.1999
%Y A003188 Same sequence in binary: A014550; bisection: A048724. Cf. A038554, A048641, A048642.
%K A003188 nonn,nice,easy
%O A003188 0,3
%A A003188 njas
%E A003188 More terms from Larry Reeves (Larryr@acm.org), Sep 05 2000
%I A006042 M2251
%S A006042 1,3,2,6,7,5,4,13,12,14,15,11,10,8,9,24,25,27,26,30,31,29,28,21,20,22,
%T A006042 23,19,18,16,17,52,53,55,54,50,51,49,48,57,56,58,59,63,62,60,61,44,45
%N A006042 Nim-squares.
%D A006042 J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%H A006042 Index entries for sequences related to Nim-multiplication
%K A006042 nonn,nice,easy
%O A006042 1,2
%A A006042 njas
%I A026137
%S A026137 1,3,2,6,7,9,4,12,5,15,16,18,19,21,8,24,25,27,10,30,11,33,34,
%T A026137 36,13,39,14,42,43,45,46,48,17,51,52,54,55,57,20,60,61,63,22,
%U A026137 66,23,69,70,72,73,75,26,78,79,81,28,84,29,87,88,90,31,93,32
%N A026137 a(n) = position of n in A026136.
%K A026137 nonn
%O A026137 1,2
%A A026137 Clark Kimberling, ck6@cedar.evansville.edu
%I A026173
%S A026173 1,3,2,6,7,9,4,12,5,15,16,18,19,21,8,24,25,27,10,30,11,33,34,
%T A026173 36,13,39,14,42,43,45,46,48,17,51,52,54,55,57,20,60,61,63,22,
%U A026173 66,23,69,70,72,73,75,26,78,79,81,28,84,29,87,88,90,31,93,32
%N A026173 a(n) = position of n in A026172.
%K A026173 nonn
%O A026173 1,2
%A A026173 Clark Kimberling, ck6@cedar.evansville.edu
%I A026187
%S A026187 1,3,2,6,7,9,4,12,5,15,16,18,19,21,8,24,25,27,10,30,11,33,34,
%T A026187 36,13,39,14,42,43,45,46,48,17,51,52,54,55,57,20,60,61,63,22,
%U A026187 66,23,69,70,72,73,75,26,78,79,81,28,84,29,87,88,90,31,93,32
%N A026187 a(n) = position of n in A026186.
%K A026187 nonn
%O A026187 1,2
%A A026187 Clark Kimberling, ck6@cedar.evansville.edu
%I A026211
%S A026211 1,3,2,6,7,9,4,12,5,15,16,18,19,21,8,24,25,27,10,30,11,33,34,
%T A026211 36,13,39,14,42,43,45,46,48,17,51,52,54,55,57,20,60,61,63,22,
%U A026211 66,23,69,70,72,73,75,26,78,79,81,28,84,29,87,88,90,31,93,32
%N A026211 a(n) = position of n in A026210.
%K A026211 nonn
%O A026211 1,2
%A A026211 Clark Kimberling, ck6@cedar.evansville.edu
%I A021310
%S A021310 0,0,3,2,6,7,9,7,3,8,5,6,2,0,9,1,5,0,3,2,6,7,9,7,3,8,5,6,2,0,9,1,5,
%T A021310 0,3,2,6,7,9,7,3,8,5,6,2,0,9,1,5,0,3,2,6,7,9,7,3,8,5,6,2,0,9,1,5,0,
%U A021310 3,2,6,7,9,7,3,8,5,6,2,0,9,1,5,0,3,2,6,7,9,7,3,8,5,6,2,0,9,1,5,0,3
%N A021310 Decimal expansion of 1/306.
%K A021310 nonn,cons
%O A021310 0,3
%A A021310 njas
%I A019444
%S A019444 1,3,2,6,8,4,11,5,14,16,7,19,21,9,24,10,27,29,12,32,13,35,37,15,40,
%T A019444 42,17,45,18,48,50,20,53,55,22,58,23,61,63,25,66,26,69,71,28,74,76,
%U A019444 30,79,31
%N A019444 a_1, a_2, ..., is a permutation of the positive integers such that the average of each initial segment is an integer, using the greedy algorithm to define a_n.
%C A019444 Self-inverse when considered as a permutation or function, i.e. a(a(n)) = n. - Howard A. Landman (howard@polyamory.org), Sep 25 2001
%H A019444 Problem of the week, Web site - problem 818
%H A019444 Index entries for sequences that are permutations of the natural numbers
%Y A019444 Cf. A019445, A019446. A019444 = A002251 + 1.
%K A019444 nonn
%O A019444 1,2
%A A019444 rkg, Tom Halverson (halverson@macalester.edu)
%I A052616
%S A052616 3,2,6,12,72,240,2160,10080,120960,725760,10886400,79833600,
%T A052616 1437004800,12454041600,261534873600,2615348736000,62768369664000,
%U A052616 711374856192000,19207121117184000,243290200817664000
%N A052616 A simple regular expression in a labeled universe.
%H A052616 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 561
%F A052616 E.g.f.: -(2*x+3)/(-1+x^2)
%F A052616 Recurrence: {a(1)=2,a(0)=3,(-2-n^2-3*n)*a(n)+a(n+2)}
%F A052616 Sum(1/2*(3*_alpha+2)*_alpha^(-1-n),_alpha=RootOf(-1+_Z^2))*n!
%p A052616 spec:= [S,{S=Union(Sequence(Z),Sequence(Z),Sequence(Prod(Z,Z)))},labelled]: seq(combstruct[count](spec,size=n),n=0..20);
%K A052616 easy,nonn
%O A052616 0,1
%A A052616 encyclopedia@pommard.inria.fr, Jan 25 2000
%I A007812
%S A007812 0,1,3,2,6,17,33,56,72,88,114,140,160,190,211,250,290,322,356,404,438,474
%N A007812 n-node Steinhaus graphs whose complements have at least one cut-vertex.
%D A007812 Cf. W. Dymacek, T. Whaley, Generating strings for bipartite Steinhaus graphs, Discrete Math. 141 (1995), pages 97-107.
%D A007812 Cf. W.M. Dymacek, M. Koerlin, T. Whaley, A survey of Steinhaus graphs, Proc. 8th Quadrennial International Conf. on Graph Theory, Combinatorics, Algorithms and Application, Kalamazoo, Mich. 1996, pages 313-323, Vol. I.
%K A007812 nonn
%O A007812 1,3
%A A007812 Wayne M. Dymacek [ DYMACEK@fs.sciences.WLU.EDU ]
%I A025240
%S A025240 3,2,6,22,90,394,1806,8558,41586,206098,1037718,5293446,27297738,142078746,
%T A025240 745387038,3937603038,20927156706,111818026018,600318853926,3236724317174,
%U A025240 17518619320890,95149655201962,518431875418926,2832923350929742
%N A025240 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.
%F A025240 G.f.: (1+3*x-sqrt(1-6*x+x^2))/2 - Michael Somos, June 8, 2000.
%o A025240 (PARI) a(n)=polcoeff((1+3*x-sqrt(1-6*x+x^2+x*O(x^n)))/2,n)
%Y A025240 Essentially same as A006318.
%K A025240 nonn
%O A025240 1,1
%A A025240 Clark Kimberling (ck6@cedar.evansville.edu)
%I A018864
%S A018864 0,3,2,6,87,1,5,41,176,9,117,4,13,94,175,8,26,89,152,215,3,30,84,129,174,
%T A018864 219,7,16,52,88,133,169,205,241,268,2,38,65,101,128,155,182,209,236,263,
%U A018864 6,15,42,69,87,114,141,159,177,204,222,240,267,285,10,28,46,64,82,100
%N A018864 6^a(n) is smallest power of 6 beginning with n.
%K A018864 nonn,base
%O A018864 1,2
%A A018864 dww
%I A019951
%S A019951 1,3,2,7,0,4,4,8,2,1,6,2,0,4,1,0,0,3,7,1,5,9,4,7,2,5,7,4,0,8,6,9,3,
%T A019951 2,4,1,9,9,0,6,0,4,1,2,9,5,5,8,7,6,2,3,0,1,6,2,0,7,7,3,5,6,8,2,5,1,
%U A019951 5,9,1,6,3,4,0,3,0,2,6,0,2,8,9,8,6,9,2,4,9,6,3,3,6,7,6,5,4,3,1,2,3
%N A019951 Decimal expansion of tangent of 53 degrees.
%K A019951 nonn,cons
%O A019951 1,2
%A A019951 njas
%I A019971
%S A019971 3,2,7,0,8,5,2,6,1,8,4,8,4,1,4,0,8,6,5,3,0,8,8,5,6,2,5,7,3,0,5,4,1,
%T A019971 0,7,7,7,1,0,5,9,4,2,6,8,4,3,1,8,8,1,0,7,0,3,6,4,0,0,8,8,0,3,4,8,2,
%U A019971 3,6,6,1,1,6,1,0,0,9,2,6,7,9,9,4,3,4,1,5,8,5,5,4,2,5,1,2,0,4,8,2,0
%N A019971 Decimal expansion of tangent of 73 degrees.
%K A019971 nonn,cons
%O A019971 1,1
%A A019971 njas
%I A010606
%S A010606 3,2,7,1,0,6,6,3,1,0,1,8,8,5,8,9,7,2,8,2,2,4,8,0,6,9,0,2,3,9,2,5,3,
%T A010606 1,3,4,4,0,9,8,9,0,3,1,4,7,7,7,8,9,0,5,8,1,9,6,4,4,5,6,0,1,0,7,8,6,
%U A010606 5,2,0,0,3,9,4,4,4,5,8,8,8,3,1,7,9,5,8,6,1,2,7,0,9,0,0,7,6,6,3,1,6
%N A010606 Decimal expansion of cube root of 35.
%K A010606 nonn,cons
%O A010606 1,1
%A A010606 njas
%I A010760
%S A010760 1,1,1,1,3,2,7,1,7,6,31,4,63,14,15,1,15,14,511,12
%N A010760 Maximal cycle length for a ring of n binary cellular automata.
%H A010760 Index entries for sequences related to cellular automata
%K A010760 nonn
%O A010760 1,5
%A A010760 gandalf@hrn.office.ssi.net (Jim Ausfahl)
%I A057020
%S A057020 1,3,2,7,3,3,4,15,13,9,6,14,7,6,6,31,9,13,10,7,8,9,12,15,31,21,10,28,
%T A057020 15,9,16,21,12,27,12,91,19,15,14,45,21,12,22,14,13,18,24,62,19,31,18,
%U A057020 49,27,15,18,15,20,45,30,14,31,24,52,127,21,18,34,21,24,18
%N A057020 Numerator of (sum of factors of n / number of factors of n).
%F A057020 a(n) =A057021(n)*A000203(n)/A000005(n) =A000203(n)/A009205(n) =(A057022(n)+A054025(n)/A000005(n))*A057021(n)
%e A057020 a(12)=14 since the 6 factors of 12 are 1, 2, 3, 4, 6 and 12, and 1+2+3+4+6+12=28 and 28/6=14/3
%Y A057020 Cf. A000005, A000203, A009205, A054025, A057021 (denominator), A057022.
%K A057020 frac,nonn
%O A057020 1,2
%A A057020 Henry Bottomley (se16@btinternet.com), Jul 21 2000
%I A011772
%S A011772 1,3,2,7,4,3,6,15,8,4,10,8,12,7,5,31,16,8,18,15,6,11,22,15,24,12,26,
%T A011772 7,28,15,30,63,11,16,14,8,36,19,12,15,40,20,42,32,9,23,46,32,48,24,
%U A011772 17,39,52,27,10,48,18,28,58,15
%N A011772 Pseudo-Smarandache numbers: a(n) = smallest number m such that m(m+1)/2 is divisible by n.
%D A011772 K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996.
%D A011772 C. Ashbacher, The Pseudo-Smarandache Function and the Classical Functions of Number Theory, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 79-82.
%H A011772 M. L. Perez et al., eds., Smarandache Notions Journal
%H A011772 E. W. Weisstein, Link to a section of The World of Mathematics.
%K A011772 nonn,easy,nice
%O A011772 1,2
%A A011772 Kenichiro Kashihara (Univxiq@aol.com)
%I A060451
%S A060451 1,3,2,7,4,3,15,5,5,4,31,9,6,6,5,63,13,7,7,7,6,127,19,11,8,8,8,7,
%T A060451 255
%N A060451 Triangle T(m,R) (1 <= R <= m) giving shortest possible length n of an [n,n-m] binary linear code with covering radius R.
%D A060451 G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 202.
%H A060451 Index entries for sequences related to covering codes
%e A060451 1; 3,2; 7,4,3; 15,5,5,4; 31,9,6,6,5; ...
%Y A060451 Cf. A060450.
%K A060451 nonn,tabl,nice,hard
%O A060451 1,2
%A A060451 njas, Apr 08 2001
%E A060451 The next entry, T(8,2), is 25 or 26.
%I A059029
%S A059029 3,2,7,4,11,6,15,8,19,10,23,12,27,14,31,16,35,18,39,20,43,22,47,24,51,
%T A059029 26,55,28,59,30,63,32,67,34,71,36,75,38,79,40,83,42,87,44,91,46,95,48,
%U A059029 99,50,103,52,107,54,111,56,115,58,119,60,123,62,127,64,131
%N A059029 Third main diagonal of A059026: a(n) = B(n+2,n) = lcm(n+2,n)/(n+2) + lcm(n+2,n)/n - 1 for all n >= 1.
%p A059029 B := (n,m) -> lcm(n,m)/n + lcm(n,m)/m - 1: seq(B(m+2,m),m=1..90);
%Y A059029 Cf. A059026, A059030, A059031.
%K A059029 nonn,easy
%O A059029 1,1
%A A059029 Asher Auel (asher.auel@reed.edu) 15 dec 2000
%I A056434
%S A056434 0,1,1,3,2,7,4,13,10,25,14,87,36,129,183,356,260,1345,804,3759,4433,
%T A056434 9757,8384,47461,33780,108329,138766,408375,331184,2251815,1155734
%N A056434 Step cyclic shifted sequence structures using exactly two different characters.
%C A056434 See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent. Permuting the characters will not change the structure.
%D A056434 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%F A056434 A056429(n)-1.
%Y A056434 Cf. A056415.
%K A056434 nonn
%O A056434 1,4
%A A056434 Marks R. Nester (nesterm@qfri1.se2.dpi.qld.gov.au)
%I A011384
%S A011384 1,3,2,7,5,3,1,6,7,4,8,8,8,5,1,9,1,9,0,6,4,3,2,5,6,8,8,6,0,2,1,2,0,
%T A011384 7,5,2,6,0,0,9,7,6,3,0,6,0,7,9,4,6,8,4,6,2,9,5,7,8,4,6,7,2,9,4,2,3,
%U A011384 2,4,2,5,3,6,0,6,5,4,6,5,6,3,3,1,3,2,3,7,6,8,9,4,3,5,9,3,7,2,3,4,6
%N A011384 Decimal expansion of 10th root of 17.
%K A011384 nonn,cons
%O A011384 1,2
%A A011384 njas
%I A059894
%S A059894 1,3,2,7,5,6,4,15,11,13,9,14,10,12,8,31,23,27,19,29,21,25,17,30,22,26,
%T A059894 18,28,20,24,16,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49,33,62,46,
%U A059894 54,38,58,42,50,34,60,44,52,36,56,40,48,32,127,95,111,79,119,87,103,71
%N A059894 Complement and reverse the order of all but the most significant bits in binary expansion of n. n = 1ab..yz -> 1ZY..BA = a(n), where A=1-a, B=1-b,..
%C A059894 A self-inverse permutation. Also a(n)=A054429(A059893(n))=A059893(A054429(n)).
%e A059894 a(9)=a(1001)=1011=11.
%Y A059894 {A000027, A054429, A059893, A059894} form a 4-group.
%K A059894 base,easy,nonn
%O A059894 1,2
%A A059894 Marc Le Brun (mlb@well.com), Feb 06 2001
%I A006921 M2252
%S A006921 1,1,3,2,7,5,13,8,29,21,55,34,115,81,209,128,465,337,883,546,1847,1301,
%T A006921 3357,2056,7437,5381,14087,8706,29443,20737,53505,32768,119041,86273,
%U A006921 226051,139778,472839,333061,859405,526344,1903901,1377557,3606327
%N A006921 Diagonals of Pascal's triangle mod 2 interpreted as binary numbers.
%D A006921 B. R. Hodgson, On some number sequences related to the parity of binomial coefficients, Fib. Quart., 30 (1992), 35-47.
%K A006921 nonn,easy
%O A006921 0,3
%A A006921 njas
%I A014693
%S A014693 3,2,7,5,14,10,21,15,28,24,37,31,48,36,55,45,68,52,77,61,84,68,95,77,
%T A014693 110,88,117,93,124,98,143,115,154,122,167,133,176,144,187,153,200,160,
%U A014693 213,171,220,176,235,199,252,204,259,213,268,224,285,235,298,242,307
%N A014693 In sequence of prime numbers add 1 to 1st number, 3rd number, 5th number, ... then subtract 1 from 2nd number, 4th number, 6th number and so on.
%K A014693 nonn,easy
%O A014693 1,1
%A A014693 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A014693 More terms from James A. Sellers (sellersj@math.psu.edu)
%I A033318
%S A033318 3,2,7,5,23,10,47,17,79,26,119,21,167,50,176,65,287,82,359,57,88,122,
%T A033318 527,28,623,170,727,111,839,44,959,257,216,290,1035,183,1367,362,19,
%U A033318 77,1679,22,1847,273,208,530,2207,456,2399,626,128,381,2807,730,696
%N A033318 Value of D corresponding to smallest solution of Pell equation x^2-Dy^2=1 being y = 1, 2, ...
%H A033318 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A033318 Cf. A033317.
%K A033318 nonn
%O A033318 0,1
%A A033318 Eric W. Weisstein (eric@weisstein.com)
%I A006068 M2253
%S A006068 0,1,3,2,7,6,4,5,15,14,12,13,8,9,11,10,31,30,28,29,24,25,27,26,16,17,
%T A006068 19,18,23,22,20,21,63,62,60,61,56,57,59,58,48,49,51,50,55,54,52,53,32,
%U A006068 33,35,34,39,38,36,37,47,46,44,45,40,41,43,42,127,126,124,125,120,121
%N A006068 a(n) is Gray-coded into n.
%C A006068 Equivalently, if binary expansion of n has m bits (say), compute derivative of n (A038554), getting sequence n' of length m-1; sort on n'.
%C A006068 Inverse of sequence A003188 considered as a permutation of the non-negative integers, i.e. A006068(A003188(n)) = n = A003188(A006068(n)). - Howard A. Landman (howard@polyamory.org), Sep 25 2001
%D A006068 M. Gardner, Mathematical Games, Sci. Amer. Vol. 227 (No. 2, Feb. 1972), p. 107.
%D A006068 M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 15.
%H A006068 Index entries for sequences that are permutations of the natural numbers
%F A006068 a(n) =2*a(ceiling[(n+1)/2])+A010060(n-1). If 3*2^(k-1) < n <= 2^(k+1), a(n)=2^(k+1)-1-a(n-2^k); if 2^(k+1) < n <= 3*2^k, a(n)=a(n-2^k)+2^k.
%e A006068 The first few values of n' are -,-,1,0,10,11,01,00,100,101,111,110,010,011,001,000,... (for n=0..15) and to put these in lexicographic order we must take n in the order 0,1,3,2,7,6,4,5,15,14,12,13,8,9,11,10,...
%Y A006068 Cf. A038554, A005811, A003188, A014550, A003100.
%K A006068 nonn,easy,nice
%O A006068 0,3
%A A006068 njas
%E A006068 Formula and more terms from Henry Bottomley (se16@btinternet.com), Jan 10 2001
%I A054429
%S A054429 1,3,2,7,6,5,4,15,14,13,12,11,10,9,8,31,30,29,28,27,26,25,24,23,22,21,
%T A054429 20,19,18,17,16,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,
%U A054429 45,44,43,42,41,40,39,38,37,36,35,34,33,32,127,126,125,124,123,122,121
%N A054429 Simple self-inverse permutation of natural numbers: List each block of 2^n numbers (from 2^n to 2^(n+1) - 1) in reverse order.
%H A054429 Index entries for sequences that are permutations of the natural numbers
%F A054429 a(n) = ReflectBinTreePermutation(n)
%p A054429 ReflectBinTreePermutation := n -> (((3*(2^floor_log_2(n)))-n)-1); # floor_log_2(x) gives [log2(x)], but floor(log[2](x)) is not healthy in Maple, so use this:
%p A054429 floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi: nn := floor(nn/2); od; end;
%Y A054429 See also A054424, A054430.
%Y A054429 {A000027, A054429, A059893, A059894} form a 4-group.
%K A054429 nonn,easy
%O A054429 1,2
%A A054429 Antti.Karttunen@iki.fi (karttu@megabaud.fi)
%I A057502
%S A057502 0,1,3,2,7,6,8,4,5,17,16,18,14,15,20,19,21,9,10,22,11,12,13,45,44,46,
%T A057502 42,43,48,47,49,37,38,50,39,40,41,54,53,55,51,52,57,56,58,23,24,59,25,
%U A057502 26,27,61,60,62,28,29,63,30,31,32,64,33,34,35,36,129,128,130,126,127
%N A057502 Permutation of natural numbers: rotations of non-crossing handshakes encoded by A014486. (to opposite direction of A057501).
%C A057502 In A057501 and A057502, the cycles between A014138[n-1]+1-th and A014138[n]th term partition A000108[n] objects encoded by the corresponding terms of A014486 into A002995[n+1] equivalence classes of planar trees, thus the latter sequence can be produced also with Maple procedure RotHandshakesPermutationCycleCounts given below.
%H A057502 Index entries for sequences that are permutations of the natural numbers
%p A057502 map(CatalanRankGlobal,map(RotateHandshakesR,A014486));
%p A057502 RotateHandshakesR := n -> pars2binexp(deepreverse(RotateHandshakesP(deepreverse(binexp2pars(n)))));
%p A057502 deepreverse:=proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else [op(deepreverse(cdr(a))), deepreverse(a[1])]; fi; end;
%p A057502 with(group); CountCycles := b -> (nops(convert(b,'disjcyc')) + (nops(b)-convert(map(nops,convert(b,'disjcyc')),`+`)));
%p A057502 RotHandshakesPermutationCycleCounts := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,RotateHandshakes(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
%p A057502 # For other procedures, follow A057501.
%Y A057502 Cf. A057507, A057513. Inverse: A057501.
%K A057502 nonn
%O A057502 0,3
%A A057502 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Sep 03 2000
%I A057504
%S A057504 0,1,3,2,7,6,8,5,4,17,16,18,15,14,20,19,21,12,11,22,13,10,9,45,44,46,
%T A057504 43,42,48,47,49,40,39,50,41,38,37,54,53,55,52,51,57,56,58,31,30,59,32,
%U A057504 29,28,61,60,62,34,33,63,35,26,25,64,36,27,24,23,129,128,130,127,126
%N A057504 Inverse permutation of A057503.
%H A057504 Index entries for sequences that are permutations of the natural numbers
%p A057504 a(n) = nthmember(n,A057503) # nthmember given in A054426.
%K A057504 nonn
%O A057504 0,3
%A A057504 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Sep 03 2000
%I A058646
%S A058646 1,1,3,2,7,6,12,10,21,22,36,36,59
%N A058646 McKay-Thompson series of class 36C for Monster.
%D A058646 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058646 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058646 nonn
%O A058646 -1,3
%A A058646 njas, Nov 27, 2000
%I A014841
%S A014841 0,3,2,7,6,13,13,18,19,31,24,37,40,48,48,65,59,79,76,85,93,117,105,
%T A014841 121,132,148,143,176,163,193,191,208,226,250,225,262,277,302,290,332,
%U A014841 320,359,363,376,394,444,419,455,462,491,495,551,540,577,564,601,625
%N A014841 Sum modulo the base of all the digits of n in every base from 2 to n-1.
%K A014841 nonn
%O A014841 3,2
%A A014841 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A056476
%S A056476 1,0,1,1,3,2,7,6,14,12,31,27,63,56,123,120,255,238,511,495,1015,992,
%T A056476 2047,2010,4092,4032,8176,8127,16383,16242,32767
%N A056476 Primitive (aperiodic) palindromic structures using a maximum of two different characters.
%C A056476 Permuting the characters will not change the structure.
%D A056476 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%F A056476 Sum mobius(d)*A016116(n/d-1) where d|n.
%Y A056476 Cf. A056458.
%K A056476 nonn
%O A056476 1,5
%A A056476 Marks R. Nester (nesterm@qfri1.se2.dpi.qld.gov.au)
%I A056481
%S A056481 0,0,1,1,3,2,7,6,14,12,31,27,63,56,123,120,255,238,511,495,1015,992,
%T A056481 2047,2010,4092,4032,8176,8127,16383,16242,32767
%N A056481 Primitive (aperiodic) palindromic structures using exactly two different characters.
%C A056481 Permuting the characters will not change the structure. Identical to A056476 for n>1.
%D A056481 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%F A056481 A056476(n)-A000007(n-1).
%Y A056481 Cf. A056463.
%K A056481 nonn
%O A056481 1,5
%A A056481 Marks R. Nester (nesterm@qfri1.se2.dpi.qld.gov.au)
%I A005213 M2254
%S A005213 0,0,1,1,3,2,7,6,19,16,51,45,141,126
%N A005213 Interval schemes.
%D A005213 Hanlon, Phil; Counting interval graphs. Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.
%K A005213 nonn
%O A005213 1,5
%A A005213 njas
%I A016603
%S A016603 3,2,7,7,1,4,4,7,3,2,9,9,2,1,7,6,5,2,4,7,2,7,2,3,7,0,1,7,5,7,0,8,8,
%T A016603 1,2,0,2,2,8,4,4,7,7,6,1,8,5,5,0,9,6,2,3,4,8,9,2,7,4,4,9,4,6,0,5,2,
%U A016603 4,6,1,1,0,1,1,9,7,3,7,9,5,1,4,2,9,2,8,7,9,6,6,1,7,4,8,9,5,0,2,5,4
%N A016603 Decimal expansion of ln(53/2).
%K A016603 nonn,cons
%O A016603 1,1
%A A016603 njas
%I A054183
%S A054183 1,0,1,1,3,2,7,7,16,19,43,58,121,182,357,603,1161,2036,3913,7131,
%T A054183 13639,25438,48733,92135,176902,337472,649514,1246672,2405235,4636007,
%U A054183 8964799,17334189,33588189,65106900,126390021,245490129,477353375
%N A054183 Mobius transform of A000011 (starting at term 0).
%K A054183 nonn
%O A054183 0,5
%A A054183 njas, Apr 29 2000
%I A057501
%S A057501 0,1,3,2,7,8,5,4,6,17,18,20,21,22,12,13,10,9,11,15,14,16,19,45,46,48,
%T A057501 49,50,54,55,57,58,59,61,62,63,64,31,32,34,35,36,26,27,24,23,25,29,28,
%U A057501 30,33,40,41,38,37,39,43,42,44,47,52,51,53,56,60,129,130,132,133,134
%N A057501 Permutation of natural numbers: rotations of non-crossing handshakes encoded by A014486.
%H A057501 Index entries for sequences that are permutations of the natural numbers
%p A057501 map(CatalanRankGlobal,map(RotateHandshakes,A014486));
%p A057501 RotateHandshakes := n -> pars2binexp(RotateHandshakesP(binexp2pars(n)));
%p A057501 RotateHandshakesP := h -> `if`((0 = nops(h)),h,[op(car(h)),cdr(h)]); # This does the trick! In Lisp: (defun RotateHandshakesP (h) (append (car h) (list (cdr h))))
%p A057501 car:=proc(a) if 0 = nops(a) then ([]) else (op(1,a)): fi: end: # The name is from Lisp, takes the first element (head) of the list.
%p A057501 cdr:=proc(a) if 0 = nops(a) then ([]) else (a[2..nops(a)]): fi: end: # As well. Takes the rest (the tail) of the list.
%p A057501 PeelNextBalSubSeq := proc(nn) local n,z,c; if(0 = nn) then RETURN(0); fi; n := nn; c := 0; z := 0; while(1 = 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); if(c >= 0) then RETURN((z - 2^(floor_log_2(z)))/2); fi; od; end;
%p A057501 RestBalSubSeq := proc(nn) local n,z,c; n := nn; c := 0; while(1 = 1) do c := c + (-1)^n; n := floor(n/2); if(c >= 0) then break; fi; od; z := 0; c := -1; while(1 = 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); if(c >= 0) then RETURN(z/2); fi; od; end;
%p A057501 pars2binexp := proc(p) local e,s,w,x; if(0 = nops(p)) then RETURN(0); fi; e := 0; for s in p do x := pars2binexp(s); w := floor_log_2(x); e := e * 2^(w+3) + 2^(w+2) + 2*x; od; RETURN(e); end;
%p A057501 binexp2pars := proc(n) option remember; `if`((0 = n),[],binexp2parsR(binrev(n))); end;
%p A057501 binexp2parsR := n -> [binexp2pars(PeelNextBalSubSeq(n)),op(binexp2pars(RestBalSubSeq(n)))];
%p A057501 # Procedure CatalanRankGlobal given in A057117, other missing ones in A038776.
%Y A057501 Inverse permutation: A057502. Max cycle lengths: A057543. Cf. A057503, A057505, A057508, A057509, A057511, A057517, A057161.
%K A057501 nonn
%O A057501 0,3
%A A057501 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Sep 03 2000
%I A057161
%S A057161 0,1,3,2,7,8,5,6,4,17,18,20,21,22,12,13,15,16,19,10,11,14,9,45,46,48,
%T A057161 49,50,54,55,57,58,59,61,62,63,64,31,32,34,35,36,40,41,43,44,47,52,53,
%U A057161 56,60,26,27,29,30,33,38,39,42,51,24,25,28,37,23,129,130,132,133,134
%N A057161 Permutation of natural numbers: rotations of the rooted plane binary trees (or triangularizations of polygons) encoded by A014486.
%H A057161 Illustration of triangulations of polygons.
%H A057161 Index entries for sequences that are permutations of the natural numbers
%p A057161 a(n) = CatalanRankGlobal(RotateBinTree(A014486[n]))
%p A057161 CatalanRankGlobal given in A057117 and the other Maple procedures in A038776.
%p A057161 NextSubBinTree := proc(nn) local n,z,c; n := nn; c := 0; z := 0; while(c < 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); od; RETURN(z); end;
%p A057161 BinTreeLeftBranch := n -> NextSubBinTree(floor(n/2));
%p A057161 BinTreeRightBranch := n -> NextSubBinTree(floor(n/(2^(1+binwidth(BinTreeLeftBranch(n))))));
%p A057161 RotateBinTree := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := BinTreeRightBranch(n); z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;
%Y A057161 Inverse permutation: A057162, Cf. also A057163, A057164, A057501, A057505. Max cycle lengths: A057544.
%K A057161 nonn
%O A057161 0,3
%A A057161 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Aug 18 2000
%I A021309
%S A021309 0,0,3,2,7,8,6,8,8,5,2,4,5,9,0,1,6,3,9,3,4,4,2,6,2,2,9,5,0,8,1,9,6,
%T A021309 7,2,1,3,1,1,4,7,5,4,0,9,8,3,6,0,6,5,5,7,3,7,7,0,4,9,1,8,0,3,2,7,8,
%U A021309 6,8,8,5,2,4,5,9,0,1,6,3,9,3,4,4,2,6,2,2,9,5,0,8,1,9,6,7,2,1,3,1,1
%N A021309 Decimal expansion of 1/305.
%K A021309 nonn,cons
%O A021309 0,3
%A A021309 njas
%I A054170
%S A054170 1,0,1,1,3,2,7,8,18,26,55,89,179,308,591,1086,2067,3834,7315,13767,
%T A054170 26263,49884,95419,182260,349712,670912,1290852,2485217,4794087,
%U A054170 9255772,17896831,34635738,67110875,130148520,252648981,490849470
%N A054170 Mobius transform of A000013 (starting at term 0).
%K A054170 nonn
%O A054170 0,5
%A A054170 njas, Apr 29 2000
%I A026136
%S A026136 1,3,2,7,9,4,5,15,6,19,21,8,25,27,10,11,33,12,13,39,14,43,45,
%T A026136 16,17,51,18,55,57,20,61,63,22,23,69,24,73,75,26,79,81,28,29,
%U A026136 87,30,31,93,32,97,99,34,35,105,36,37,111,38,115,117,40,41,123
%N A026136 For n >= 2, let L=n-[ n/2 ], R=n+[ n/2 ]; then a(L)=n if a(L) not yet defined, else a(R)=n; thus |a(n)-n|=[ (1/2)*a(n) ].
%K A026136 nonn
%O A026136 1,2
%A A026136 Clark Kimberling, ck6@cedar.evansville.edu
%I A026172
%S A026172 1,3,2,7,9,4,5,15,6,19,21,8,25,27,10,11,33,12,13,39,14,43,45,
%T A026172 16,17,51,18,55,57,20,61,63,22,23,69,24,73,75,26,79,81,28,29,
%U A026172 87,30,31,93,32,97,99,34,35,105,36,37,111,38,115,117,40,41,123
%N A026172 For n >= 2, let h={n/2 ], L=n-h, R=n+h; then a(R)=n if n even or a(L) already defined, else a(L)=n.
%K A026172 nonn
%O A026172 1,2
%A A026172 Clark Kimberling, ck6@cedar.evansville.edu
%I A026186
%S A026186 1,3,2,7,9,4,5,15,6,19,21,8,25,27,10,11,33,12,13,39,14,43,45,
%T A026186 16,17,51,18,55,57,20,61,63,22,23,69,24,73,75,26,79,81,28,29,
%U A026186 87,30,31,93,32,97,99,34,35,105,36,37,111,38,115,117,40,41,123
%N A026186 a(n) = (1/3)*(s(n) + 2), where s(n) is the n-th number congruent to 1 mod 3 in A026136.
%K A026186 nonn
%O A026186 1,2
%A A026186 Clark Kimberling, ck6@cedar.evansville.edu
%I A026210
%S A026210 1,3,2,7,9,4,5,15,6,19,21,8,25,27,10,11,33,12,13,39,14,43,45,
%T A026210 16,17,51,18,55,57,20,61,63,22,23,69,24,73,75,26,79,81,28,29,
%U A026210 87,30,31,93,32,97,99,34,35,105,36,37,111,38,115,117,40,41,123
%N A026210 a(n) = (1/3)*(s(n)+2), where s(n) is the n-th number congruent to 1 mod 3 in A026172.
%K A026210 nonn
%O A026210 1,2
%A A026210 Clark Kimberling, ck6@cedar.evansville.edu
%I A018891
%S A018891 0,0,1,0,1,0,1,1,1,3,2,7,9,17,47
%N A018891 Positive knots with n crossings.
%H A018891 D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B 393, No.3-4, 403-412 (1997).
%K A018891 nonn,nice
%O A018891 1,10
%A A018891 David Broadhurst (D.Broadhurst@open.ac.uk)
%I A034423
%S A034423 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3,
%T A034423 2,7,9,23,30,65,95,179,272,487,751,1291,2020,3367,5308,8719,13777,
%U A034423 22420,35613
%N A034423 Multiplicity of highest weight (or singular) vectors associated with character chi_35 of Monster module.
%D A034423 K. Harada and M. L. Lung, Modular forms associated with the Monster module, pp. 59-83 of The Monster and Lie Algebras, de Gruyter, 1998.
%K A034423 nonn
%O A034423 1,33
%A A034423 njas
%I A053440
%S A053440 1,3,2,7,12,6,15,50,60,24,31,180,390,360,120,63,602,2100,3360,2520,720,
%T A053440 127,1932,10206,25200,31920,20160,5040,255,6050,46620,166824,317520,
%U A053440 332640,181440,40320,511,18660,204630,1020600,2739240,4233600,3780000
%N A053440 Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n, k) is the number of ascending chains of length k+1 of non-empty subsets of the set {1, 2, ..., n+1}.
%F A053440 T(0, k) = delta(0, k), T(n, k) = delta(0, k) + (k+1)(T(n-1, k-1) + (k+2)T(n-1, k)).
%Y A053440 Cf. A028246.
%K A053440 nonn,easy,tabl,nice
%O A053440 0,2
%A A053440 Rob Arthan (rda@lemma-one.com), Jan 12 2000
%E A053440 More terms from James A. Sellers (sellersj@math.psu.edu), Jan 14 2000
%I A052546
%S A052546 1,0,1,3,2,7,13,18,41,71,122,239,421,762,1417,2543,4642,8495,15389,
%T A052546 28082,51177,93047,169610,308847,562197,1024170,1864841,3395711,
%U A052546 6184498,11261551,20507789,37346914,68008809,123848199,225535258
%N A052546 A simple regular expression.
%H A052546 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 482
%F A052546 G.f.: -(-1+x)/(1-x-2*x^3+2*x^4-x^2)
%F A052546 Recurrence: {a(1)=0,a(0)=1,a(2)=1,a(3)=3,2*a(n)-2*a(n+1)-a(n+2)-a(n+3)+a(n+4)}
%F A052546 Sum(-1/353*(-18-106*_alpha+33*_alpha^2+28*_alpha^3)*_alpha^(-1-n),_alpha=RootOf(1-_Z-2*_Z^3+2*_Z^4-_Z^2))
%p A052546 spec:= [S,{S=Sequence(Prod(Z,Z,Union(Z,Z,Sequence(Z))))},unlabelled]: seq(combstruct[count](spec,size=n),n=0..20);
%K A052546 easy,nonn
%O A052546 0,4
%A A052546 encyclopedia@pommard.inria.fr, Jan 25 2000
%E A052546 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 06 2000
%I A049968
%S A049968 1,3,2,7,15,29,59,131,306,554,1109,2231,4506,9259,19321,42039,98893,
%T A049968 178466,356933,713879,1427802,2855851,5712505,11428407,22871629,
%U A049968 45822830,91903700,184878269,374041241,765241606,1599515283
%N A049968 a(n)=a(1)+a(2)+...+a(n-1)+a(m), where m=2n-3-2^(p+1), and 2^pIndex entries for sequences that are permutations of the natural numbers
%p A057162 a(n) = CatalanRankGlobal(RotateBinTreeR(A014486[n])) or a(n) = A057163[A057161[A057163[n]]]
%p A057162 RotateBinTreeR := n -> ReflectBinTree(RotateBinTree(ReflectBinTree(n)));
%p A057162 with(group); RotBinTreePermutationCycleCounts := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,RotateBinTree(CatalanUnrank(n,r)))]; od; a := [op(a),(`if`((n < 2),1,nops(convert(b,'disjcyc'))))]; od; RETURN(a); end;
%Y A057162 Inverse permutation: A057161.
%K A057162 nonn
%O A057162 0,3
%A A057162 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Aug 18 2000
%I A057506
%S A057506 0,1,3,2,8,6,7,5,4,22,19,20,15,14,21,16,18,13,11,17,12,10,9,64,60,61,
%T A057506 52,51,62,53,55,41,39,54,40,38,37,63,56,57,43,42,59,47,50,36,33,48,34,
%U A057506 29,28,58,44,49,35,30,46,32,27,25,45,31,26,24,23,196,191,192,178,177
%N A057506 Inverse permutation of A057505.
%H A057506 Index entries for sequences that are permutations of the natural numbers
%p A057506 map(CatalanRankGlobal,map(RotateHandshakesD3R,A014486));
%p A057506 RotateHandshakesD3R := n -> pars2binexp(deepreverse(RotateHandshakesD3P(deepreverse(binexp2pars(n)))));
%Y A057506 Cf. A057502 (for deepreverse), A057501 (for other procedures), A057503, A057507 (cycle counts).
%K A057506 nonn
%O A057506 0,3
%A A057506 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Sep 03 2000
%I A057503
%S A057503 0,1,3,2,8,7,5,4,6,22,21,18,17,20,13,12,10,9,11,15,14,16,19,64,63,59,
%T A057503 58,62,50,49,46,45,48,55,54,57,61,36,35,32,31,34,27,26,24,23,25,29,28,
%U A057503 30,33,41,40,38,37,39,43,42,44,47,52,51,53,56,60,196,195,190,189,194
%N A057503 Permutation of natural numbers: recursive variant of A057501.
%H A057503 Index entries for sequences that are permutations of the natural numbers
%p A057503 map(CatalanRankGlobal,map(RotateHandshakesD2,A014486));
%p A057503 RotateHandshakesD2 := n -> pars2binexp(RotateHandshakesD2P(binexp2pars(n)));
%p A057503 RotateHandshakesD2P := h -> `if`((0 = nops(h)),h,[op(car(h)),RotateHandshakesD2P(cdr(h))]);
%Y A057503 Inverse permutation: A057504. Cf. A057501 (for needed procedures), A057505.
%K A057503 nonn
%O A057503 0,3
%A A057503 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Sep 03 2000
%I A057505
%S A057505 0,1,3,2,8,7,5,6,4,22,21,18,20,17,13,12,15,19,16,10,11,14,9,64,63,59,
%T A057505 62,58,50,49,55,61,57,46,48,54,45,36,35,32,34,31,41,40,52,60,56,43,47,
%U A057505 53,44,27,26,29,33,30,38,39,51,42,24,25,28,37,23,196,195,190,194,189
%N A057505 Permutation of natural numbers: deeply recursive variant of A057501.
%C A057505 Also recursive variant of A057161, can be generated with map(CatalanRankGlobal,map(DeepRotateTriangularization,A014486)); .
%H A057505 Index entries for sequences that are permutations of the natural numbers
%p A057505 map(CatalanRankGlobal,map(RotateHandshakesD3,A014486));
%p A057505 RotateHandshakesD3 := n -> pars2binexp(RotateHandshakesD3P(binexp2pars(n)));
%p A057505 RotateHandshakesD3P := h -> `if`((0 = nops(h)),h,[op(RotateHandshakesD3P(car(h))),RotateHandshakesD3P(cdr(h))]);
%p A057505 DeepRotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := DeepRotateTriangularization(BinTreeRightBranch(n))*2; z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;
%Y A057505 Inverse permutation: A057506. Cycle counts: A057507. Maximum cycle lengths: A057545. See A057501 for procedures.
%Y A057505 LCM's of all cycles: A060114.
%K A057505 nonn
%O A057505 0,3
%A A057505 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Sep 03 2000
%I A057163
%S A057163 0,1,3,2,8,7,6,5,4,22,21,20,18,17,19,16,15,13,12,14,11,10,9,64,63,62,
%T A057163 59,58,61,57,55,50,49,54,48,46,45,60,56,53,47,44,52,43,41,36,35,40,34,
%U A057163 32,31,51,42,39,33,30,38,29,27,26,37,28,25,24,23,196,195,194,190,189
%N A057163 Self-inverse permutation of natural numbers: reflections of the rooted plane binary trees and triangularizations of polygons encoded by A014486.
%C A057163 Each A000108[n] n+2 side polygon triangularizations (and the corresponding rooted binary plane trees of 2n edges) can be reflected over n+2 axes of symmetry, which all can be generated by appropriate compositions of the permutations A057161/A057162 and A057163.
%H A057163 Index entries for sequences that are permutations of the natural numbers
%e A057163 Example: a(5)=7 and a(7)=5, A014486[5] = 44 (101100 in binary), A014486[7] = 52 (110100 in binary) and these encode the following rooted plane binary trees, which are reflections of each other:
%e A057163 .....0.0...............0.0....
%e A057163 ......1...0.........0...1.....
%e A057163 ..0.....1.............1.....0.
%e A057163 .....1...................1....
%p A057163 a(n) = CatalanRankGlobal(ReflectBinTree(A014486[n])) [For other Maple procedures, follow A057161.]
%p A057163 ReflectBinTree := n -> ReflectBinTree2(n)/2; ReflectBinTree2 := n -> (`if`((0 = n),n,ReflectBinTreeAux(binrev(n))));
%p A057163 ReflectBinTreeAux := proc(n) local a,b; a := ReflectBinTree2(BinTreeLeftBranch(n)); b := ReflectBinTree2(BinTreeRightBranch(n)); RETURN((2^(binwidth(b)+binwidth(a))) + (b * (2^(binwidth(a)))) + a); end;
%Y A057163 A057123[A057163[n]] = A057164[A057123[n]] for all n.
%K A057163 nonn
%O A057163 0,3
%A A057163 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Aug 18 2000
%I A021308
%S A021308 0,0,3,2,8,9,4,7,3,6,8,4,2,1,0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,1,0,
%T A021308 5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,1,0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,
%U A021308 2,1,0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,1,0,5,2,6,3,1,5,7,8,9,4,7,3
%N A021308 Decimal expansion of 1/304.
%K A021308 nonn,cons
%O A021308 0,3
%A A021308 njas
%I A060921
%S A060921 1,3,2,8,10,3,21,38,22,4,55,130,111,40,5,144,420,474,256,65,6,377,1308,
%T A060921 1836,1324,511,98,7,987,3970,6666,6020,3130,924,140,8,2584,11822,23109,
%U A060921 25088,16435,6588,1554,192,9
%N A060921 Bisection of Fibonacci triangle A037027: odd indexed members of column sequences of A037027 (not counting leading zeros).
%C A060921 Row sums give A002450. Column sequences (without leading zeros) give for m=0..5: A001906, 2*A001870, A061182, 4*A061183, A061184, 2*A061185.
%C A060921 Companion triangle (odd indexed members) A060920.
%F A060921 a(n,m)=A037027(2*n+1-m,m).
%F A060921 a(n,m)= (2*(n-m+1)*A060920(n,m-1)+2*(2*n+1)*a(n-1,m-1))/(5*m), n >= m>0; a(n,0):= S(n,3)=A001906(n+1) with Chebyshev's S(n,x) polynomials A049310; else 0.
%F A060921 G.f. for column m >= 0: x^m*pFo(m+1,x)/(1-3*x+x^2)^(m+1), where pFo(n,x):=sum(A061177(n-1,m)*x^m,m=0..n-1) (row polynomials of signed triangle A061177).
%e A060921 {1}; {3,2}; {8,10,3}; {21,38,22,4}; ...; pFo(2,x)= 2*(1-x).
%K A060921 nonn,easy,tabl
%O A060921 0,2
%A A060921 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Apr 20 2001
%I A047946
%S A047946 3,2,8,17,48,122,323,842,2208,5777,15128,39602,103683,271442,710648,
%T A047946 1860497,4870848,12752042,33385283,87403802,228826128,599074577,
%U A047946 1568397608,4106118242,10749957123,28143753122,73681302248
%N A047946 5*F(n)^2+3*(-1)^n where F(n) are the Fibonacci numbers A000045.
%F A047946 a(n)=F(3n)/F(n), n>0; a(n)=2*a(n-1)+2*a(n-2)-a(n-3); a(n)=3a(n-1)-a(n-2)+5(-1)^n; a(n) = L(2n) + (-1)^n, where the L(n) are Lucas numbers A000032. G.f.: (3-4*x-2*x^2)/(1-2*x-2*x^2+x^3).
%o A047946 (PARI) a(n)=5*F(n)^2+3*(-1)^n; F(n)=fibonacci(n)
%Y A047946 Cf. A000045, A000032.
%K A047946 nonn,easy
%O A047946 0,1
%A A047946 John W. Layman (layman@math.vt.edu (5/21/99))
%E A047946 Entry improved by comments from Michael Somos.
%I A060481
%S A060481 1,0,1,0,3,2,9,0,28,24,93,20,315,288,1091,0,3855
%N A060481 Number of orbits of length n in map whose periodic points come from A059991.
%D A060481 V. Chothi, G. Everest, T. Ward. S-integer dynamical systems: periodic points. J. Reine Angew. Math., 489 (1997), 99-132.
%D A060481 T. Ward. Almost all S-integer dynamical systems have many periodic points. Erg. Th. Dynam. Sys. 18 (1998), 471-486.
%F A060481 If b(n) is the n-th term of A059991, then a(n)=(1/n)*\sum_{d|n}\mu(d)a(n/d)
%Y A060481 A059991.
%K A060481 easy,nonn
%O A060481 1,5
%A A060481 Thomas Ward (t.ward@uea.ac.uk)
%I A010271
%S A010271 3,2,9,1,12,1,14,1,1,1,16,1,2,1,8,1,19,1,7,1,6,2,2,2,1,
%T A010271 8,1,40,2,807,1,26,1,1,2,1,7,2,8,9,2,5,1,5,9,41,1,4,7,4,
%U A010271 1,2,72,2,1,2,158,1,5,2,1,4,1,1,2,1,736,1,1,3,13,6,1,1
%N A010271 Continued fraction for cube root of 42.
%K A010271 nonn,cofr
%O A010271 0,1
%A A010271 njas
%I A058973
%S A058973 1,3,2,9,2,4,9,9
%N A058973 First integer reached in A058972.
%K A058973 nonn,easy,more
%O A058973 1,2
%A A058973 njas, Jan 14 2001
%I A064614
%S A064614 1,3,2,9,5,6,7,27,4,15,11,18,13,21,10,81,17,12,19,45,14,33,23,54,25,39,
%T A064614 8,63,29,30,31,243,22,51,35,36,37,57,26,135,41,42,43,99,20,69,47,162,
%U A064614 49,75,34,117,53,24,55,189,38,87,59,90,61,93,28,729,65,66,67,153,46
%N A064614 Exchange 2 and 3 in the prime factorization of n.
%C A064614 A self-inverse permutation of the natural numbers.
%C A064614 a(1) = 1, a(2) = 3, a(3) = 2, a(p) = p for primes p > 3 and a(u * v) = a(u) * a(v) for u, v > 0.
%C A064614 a is a permutation of all numbers: a(a(n)) = n for all n and a(n) = n iff n = 6^k * m for k >= 0 and m > 0 with gcd(m, 6) = 1 (see A064615).
%H A064614 Index entries for sequences that are permutations of the natural numbers
%e A064614 a(15) = a(3*5) = a(3)*a(5) = 2*5 = 10; a(16) = a(2^4) = a(2)^4 = 3^4 = 81; a(17) = 17; a(18) = a(2*3^2) = a(2)*a(3^2) = 3*a(3)^2 = 3*2^2 = 12.
%Y A064614 A064615.
%K A064614 mult,nice,nonn
%O A064614 1,2
%A A064614 Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Sep 25 2001
%I A016650
%S A016650 3,2,9,5,8,3,6,8,6,6,0,0,4,3,2,9,0,7,4,1,8,5,7,3,5,7,1,0,7,6,7,5,7,
%T A016650 7,1,1,3,9,4,2,4,7,1,6,7,3,4,6,8,2,4,8,3,5,5,2,0,4,0,8,3,0,0,0,9,1,
%U A016650 2,4,8,2,8,7,9,6,5,5,8,2,6,9,0,0,6,2,0,8,4,7,2,6,4,4,4,1,1,9,6,2,6
%N A016650 Decimal expansion of ln(27).
%D A016650 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
%K A016650 nonn,cons
%O A016650 1,1
%A A016650 njas
%I A033313
%S A033313 3,2,9,5,8,3,19,10,7,649,15,4,33,17,170,9,55,197,24,5,51,26,127,9801,
%T A033313 11,1520,17,23,35,6,73,37,25,19,2049,13,3482,199,161,24335,48,7,99,50,
%U A033313 649,66249,485,89,15,151,19603,530,31,1766319049,63,8,129,65,48842,33
%N A033313 Smallest integer x satisfying Pell equation x^2-ny^2=1 for non-square n.
%H A033313 E. W. Weisstein, Link to a section of The World of Mathematics.
%K A033313 nonn,frac
%O A033313 0,1
%A A033313 Eric W. Weisstein (eric@weisstein.com)
%I A019778
%S A019778 3,2,9,7,4,4,2,5,4,1,4,0,0,2,5,6,2,9,3,6,9,7,3,0,1,5,7,5,6,2,8,3,2,
%T A019778 7,1,4,3,3,0,7,5,5,2,2,0,1,4,2,0,2,9,6,0,2,3,1,5,0,1,5,8,6,2,3,2,8,
%U A019778 1,3,2,2,0,4,2,3,8,8,4,3,1,2,1,7,2,6,5,5,5,3,0,4,0,1,1,2,7,3,3,2,8
%N A019778 Decimal expansion of sqrt(E)/5.
%K A019778 nonn,cons
%O A019778 0,1
%A A019778 njas
%I A011323
%S A011323 1,3,2,9,7,5,4,5,4,5,6,3,9,7,8,5,9,7,2,9,0,5,2,7,3,8,7,5,1,4,8,8,1,
%T A011323 3,1,7,9,4,2,7,0,8,7,5,0,0,4,6,4,7,8,2,0,2,0,8,3,1,0,1,5,9,2,9,8,7,
%U A011323 3,8,6,4,5,4,2,5,5,2,4,3,9,5,3,0,2,7,3,6,8,5,5,4,0,6,9,6,1,1,6,4,2
%N A011323 Decimal expansion of 9th root of 13.
%K A011323 nonn,cons
%O A011323 1,2
%A A011323 njas
%I A061898
%S A061898 1,3,2,9,7,6,5,27,4,21,13,18,11,15,14,81,19,12,17,63,10,39,29,54,49,33,
%T A061898 8,45,23,42,37,243,26,57,35,36,31,51,22,189,43,30,41,117,28,87,53,162,
%U A061898 25,147,38,99,47,24,91,135,34,69,61,126,59,111,20,729,77,78,71,171,58
%N A061898 Swap each prime in factorization of n with "neighbor" prime.
%C A061898 Here "neighbor" primes are just paired in order: 2<->3, 5<->7, 11<->13, etc. Self-inverse permutation of the integers. Multiplicative.
%e A061898 a(60)=126 since 60=2^2*3*5, swapping 2<->3 and 5<->7 gives 3^2*2*7=126 (and of course then a(126)=60).
%Y A061898 A045965.
%K A061898 easy,nonn
%O A061898 1,2
%A A061898 Marc Le Brun (mlb@well.com), May 14 2001
%I A021756
%S A021756 0,0,1,3,2,9,7,8,7,2,3,4,0,4,2,5,5,3,1,9,1,4,8,9,3,6,1,7,0,2,1,2,7,
%T A021756 6,5,9,5,7,4,4,6,8,0,8,5,1,0,6,3,8,2,9,7,8,7,2,3,4,0,4,2,5,5,3,1,9,
%U A021756 1,4,8,9,3,6,1,7,0,2,1,2,7,6,5,9,5,7,4,4,6,8,0,8,5,1,0,6,3,8,2,9,7
%N A021756 Decimal expansion of 1/752.
%K A021756 nonn,cons
%O A021756 0,4
%A A021756 njas
%I A050676
%S A050676 1,3,2,9,8,11,28,6,26,20
%N A050676 Let b(n) = number of prime factors (with multiplicity) of concatenation of numbers from 1 to n; sequence gives smallest number m with b(m) = n.
%D A050676 M. Fleuren, Smarandache Factors and Reverse Factors, Smarandache Notions Journal, 10 (No. 1-2-3, Spring 1999), 5-38.
%H A050676 M. L. Perez et al., eds., Smarandache Notions Journal
%H A050676 M. Fleuren, Smarandache factors
%H A050676 C. Rivera, Primes by Listing
%H A050676 P. De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
%Y A050676 Cf. A007908, A050678, A046460.
%K A050676 nonn,base,hard
%O A050676 1,2
%A A050676 Patrick De Geest (pdg@worldofnumbers.com), Aug 1999.
%I A010372
%S A010372 1,0,1,1,3,2,9,8,35,39,159,202,802,1078,4347,6354,24894,38157,
%T A010372 148284,237541,910726,1511717,5731580,9816092,36797588,64658432,
%U A010372 240215803,431987953,1590507121,2917928218,10660307791,19910436898
%N A010372 Quartic trees (or hydrocarbons C_n H_{2n+2}) with a centroid (i.e. one unique node at minimal maximal distance from all other nodes).
%D A010372 F. Harary, Graph Theory, p. 36, for definition of centroid.
%H A010372 Index entries for sequences related to trees
%p A010372 with(combstruct): Alkyl := proc(n) combstruct[count]([ U,{U=Prod(Z,Set(U,card<=3))},unlabelled ],size=n) end:
%p A010372 centeredHC := proc(n) option remember; local f,k,z,f2,f3,f4; f := 1 + add(Alkyl(k)*z^k, k=0..iquo(n-1,2));
%p A010372 f2 := series(subs(z=z^2,f), z, n+1); f3 := series(subs(z=z^3,f), z, n+1); f4 := series(subs(z=z^4,f), z, n+1);
%p A010372 f := series(f*f3/3+f4/4+f2^2/8+f2*f^2/4+f^4/24, z, n+1); coeff(f, z, n-1) end: seq(centeredHC(n), n=1..32);
%Y A010372 Cf. A010373, A000022.
%K A010372 nonn,easy,nice
%O A010372 1,5
%A A010372 Paul.Zimmermann@inria.fr, njas
%I A038220
%S A038220 1,3,2,9,12,4,27,54,36,8,81,216,216,96,16,243,810,1080,720,240,
%T A038220 32,729,2916,4860,4320,2160,576,64,2187,10206,20412,22680,15120,
%U A038220 6048,1344,128,6561,34992,81648,108864,90720,48384,16128,3072,256
%N A038220 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
%D A038220 B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct. 1996), pp. 109-121.
%K A038220 nonn,tabl,easy
%O A038220 0,2
%A A038220 njas
%I A053151
%S A053151 1,0,3,2,9,12,31,53,116,215,446,849,1726,3320,6688,12932,25925,50297,
%T A053151 100546,195562,390257,760630,1516233,2960502,5897425,11533047,22964739,
%U A053151 44972711,89528901,175546608,349425044,685913758,1365249931,2682660933
%N A053151 Directed EG-convex polyominoes on the honeycomb lattice with given semi-perimeter.
%D A053151 Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux decales oscillants . These de Doctorat. Laboratoire de Recherche en Informatique, Universite Paris-Sud XI, France.
%D A053151 Alain Denise, Christoph Durr and Fouad Ibn-Majdoub-Hassani. Enumeration et generation aleatoire de polyominos convexes en reseau hexagonal (French) [enumeration and random generation of convex polyominoes in the honeycomb lattice] . In Proceedings of 9th Conference on Formal Power Series and Algebraic Combinatorics, pages 222-234, 1997.
%F A053151 G.f.: [x^3*((1-2*x)*(-3+x+3*x^2+2*x^3)+(1-x-x^2)*(1-4*x^2)^(1/2))]/(2*(1+x)*(1-2*x)*(1-x-x^2)*(-1+x+2*x^2+x^3))
%K A053151 nonn
%O A053151 3,3
%A A053151 Fouad IBN MAJDOUB HASSANI (Fouad.Hassani@genset.fr), Feb 28 2000
%E A053151 More terms from James A. Sellers (sellersj@math.psu.edu), Mar 01 2000
%I A053088
%S A053088 1,0,3,2,9,12,31,54,117,224,459,906,1825,3636,7287,14558,29133,58248,
%T A053088 116515,233010,466041,932060,1864143,3728262,7456549,14913072,29826171,
%U A053088 59652314,119304657,238609284,477218599,954437166,1908874365
%N A053088 a(n)=2a(n-3)+3a(n-2)
%C A053088 Growth of happy bug population in GCSE maths coursework assignment.
%F A053088 G.f.: 1/(1-3x^2-2x^3).
%K A053088 nonn,easy
%O A053088 0,3
%A A053088 Pauline Gorman (pauline@gorman65.freeserve.co.uk), Feb 26 2000
%E A053088 More terms from James A. Sellers (sellersj@math.psu.edu), Feb 28 2000 and Christian G. Bower (bowerc@usa.net), Feb 29 2000.
%I A049969
%S A049969 1,3,2,9,16,40,73,147,292,730,1386,2739,5454,10901,21795,43591,87180,
%T A049969 217950,414104,817314,1629181,3255647,6509941,13019226,26038014,
%U A049969 52075883,104151692,208303351,416606678,833213349,1666426691
%N A049969 a(n)=a(1)+a(2)+...+a(n-1)+a(m), where m=2^(p+1)+2-n, and 2^pThe point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%Y A063549 Cf. A063550.
%K A063549 nonn,nice,hard
%O A063549 3,1
%A A063549 njas, Aug 14 2001
%I A057977
%S A057977 1,1,3,2,10,5,35,14,126,42,462,132,1716,429,6435,1430,24310,4862,92378,
%T A057977 16796,352716,58786,1352078,208012,5200300,742900,20058300,2674440,
%U A057977 77558760,9694845,300540195,35357670,1166803110,129644790,4537567650
%N A057977 GCD of consecutive central binomial coefficients: a(n)=GCD[A001405(n+1),A001405(n)].
%e A057977 This GCD equals A001405(n) for the smaller odd number GCD[C[12,6],C[11,5]]=GCD[924,462]=462=C[11,5]
%Y A057977 A001405.
%K A057977 nonn
%O A057977 0,3
%A A057977 Labos E. (labos@ana1.sote.hu), Nov 13 2000
%I A056861
%S A056861 0,1,3,2,10,7,6,37,27,23,21,151,114,97,88,83
%N A056861 Triangle T(n,k) = number of element-subset partitions of {1..n} that have an increase at index k (n >= 1, 1<=k<=n).
%C A056861 Number of rises s_{i+1} > s_i in a set partition {s_1, ..., s_n} of {1, ..., n}, where s_i is the subset containing i, s(1) = 1, and s(i) <= 1 + max of previous s(j)'s.
%D A056861 W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.
%e A056861 For example {1, 2, 1, 2, 2, 3} is a set partition of {1, 2, 3, 4, 5, 6} and has 2 rises, at i = 1, i = 3, and i = 5.
%e A056861 0; 1; 3,2; 10,7,6; 37,27,23,21; 151,114,97,88,83; ...
%Y A056861 Cf. Bell numbers A000110.
%Y A056861 Cf. A056857-A056863.
%K A056861 easy,nonn,tabl,more
%O A056861 1,3
%A A056861 Winston C. Yang (winston@cs.wisc.edu), Aug 31 2000
%I A019242
%S A019242 0,3,2,10,10,19,50,86,184,436,980,2500,5890,15348,37898,96636,
%T A019242 249382,637710
%N A019242 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite STI = Stilbite Na4Ca8[ Al20Si52O144 ] . 56 H2O.
%D A019242 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019242 G. Thimm, Cycle sequences of crystal structures
%K A019242 nonn
%O A019242 3,2
%A A019242 Georg Thimm (mgeorg@ntu.edu.sg)
%I A064367
%S A064367 0,1,3,2,10,12,9,9,6,9,2,26,33,1,9,28,33,27,13,48,8,36,47,4,95,20,76,
%T A064367 62,23,4,8,117,68,25,138,64,150,43,61,10,72,156,40,12,73,51,48,41,24,
%U A064367 26,71,48,32,16,128,173,74,110,118,59,30,247,202,208,284,53,128,32,139
%N A064367 a(n) = Mod[2^n,Prime[m]], or 2^n=k*p(n)+a(n), k is integer.
%F A064367 a(n)=Mod[A000079(n),A000040(n)]
%e A064367 Below the exponent n<10000, some integers (like 5,7,14,17,19,22,..,44, etc) are not yet present among residues. Will they appear later?
%Y A064367 A000040, A000079, A015910.
%K A064367 nonn
%O A064367 1,3
%A A064367 Labos E. (labos@ana1.sote.hu), Sep 27 2001
%I A006743 M2255
%S A006743 1,0,3,2,10,14,40,74,176,358,798,1670,3626,7638,16366,34462,
%T A006743 73230,153830,324896,680514,1430336,2987310,6253712,13025954,
%U A006743 27176052,56465878,117458820,243507250,505239264,1045301486
%N A006743 Convex polygons of length 2n on honeycomb, or EG-convex polyominoes.
%D A006743 A. J. Guttmann and I. G. Enting, The number of convex polygons on the square and honeycomb lattices, J. Phys. A 21 (1988), L467-L474.
%D A006743 Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux decales oscillants . These de Doctorat. Laboratoire de Recherche en Informatique, Universite Paris-Sud XI, France.
%D A006743 Alain Denise, Christoph Durr and Fouad Ibn-Majdoub-Hassani. Enumeration et generation aleatoire de polyominos convexes en reseau hexagonal (French) [enumeration and random generation of convex polyominoes in the honeycomb lattice] . In Proceedings of 9th Conference on Formal Power Series and Algebraic Combinatorics, pages 222-234, 1997.
%F A006743 G.f.: (1-2*x+x^2-x^4-x^2*sqrt(1-4*x^2))/(1+x)^2/(1-2*x)^2, from Paul.Zimmermann@loria.fr (Paul Zimmermann)
%F A006743 a(n) = (-6*a(n-1) + 2*a(n-1)*n-33*a(n-2) + 7*a(n-2)*n + 60*a(n-3)-12*a(n-3)*n + 84*a(n-4)-16*a(n-4)*n-96*a(n-5) + 16*n*a(n-5)-96*a(n-6) + 16*n*a(n-6))/(n-3), from Simon Plouffe (plouffe@math.uqam.ca)
%K A006743 nonn,easy,nice
%O A006743 3,3
%A A006743 SP,njas
%E A006743 Additional references from Fouad IBN MAJDOUB HASSANI (Fouad.Hassani@genset.fr), Feb 28 2000.
%I A025520
%S A025520 3,2,10,15,14,13,38,146,145,144,143,142,222,221,220,219,218,217,216,215,
%T A025520 214,374,373,372,574,573,572,571,570,569,568,567,566,565,564,563,562,561,
%U A025520 560,7694,7693,7692,7691,7690,7689,7688,7687,7686,7685,7684,7683,7682
%N A025520 Least k>1 such that reverse of first n terms of A022303 repeats beginning at k-th term.
%K A025520 nonn
%O A025520 1,1
%A A025520 dww
%I A011953
%S A011953 3,2,10,15,42,68,162,280,600,1088,2214,4112,8154,15366,30052,57208,
%T A011953 111166,212982,412890,794688,1540002,2974150,5767344,11167872,
%U A011953 21680832,42073640,81787057,159005154,309503502,602667098
%N A011953 Barlow packings with group P3m1 that repeat after n layers.
%D A011953 T. J. McLarnan, The numbers of polytypes ..., Zeits. Krist. 155, 269-291 (1981).
%K A011953 nonn,easy
%O A011953 9,1
%A A011953 njas
%I A038519
%S A038519 1,0,1,3,2,10,16,28,72,120,256,528,992,2080,4096,8128,16512,32640,
%T A038519 65536,131328,261632,524800,1048576,2096128,4196352,8386560,16777216,
%U A038519 33558528,67100672,134225920,268435456,536854528,1073774592
%N A038519 Number of elements of GF(2^n) with trace 0 and subtrace 1.
%H A038519 F. Ruskey, Number of irreducible polynomials over GF(2) with given trace and subtrace
%F A038519 C(n,r+0)+C(n,r+4)+C(n,r+8)+... where r = 2 if n odd, r = 0 if n even.
%Y A038519 Cf. A038503, A038505.
%K A038519 easy,nonn
%O A038519 0,4
%A A038519 Frank Ruskey (fruskey@csr.csc.uvic.ca)
%I A034461
%S A034461 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,2,10,
%T A034461 19,46,81,179,317,636,1139,2153,3841,7090,12575,22835,40668,73558,
%U A034461 132129,241081,440946,820710,1543715
%N A034461 Multiplicity of highest weight (or singular) vectors associated with character chi_73 of Monster module.
%D A034461 K. Harada and M. L. Lung, Modular forms associated with the Monster module, pp. 59-83 of The Monster and Lie Algebras, de Gruyter, 1998.
%K A034461 nonn
%O A034461 1,31
%A A034461 njas
%I A013945
%S A013945 3,2,11,5,27,10,51,17,83,26,123,37,171,50,227,65,291,82,363,101,443,
%T A013945 122,531,145,627,170,731,197,843,226,963,257,1091,290,1227,325,1371,
%U A013945 362,1523,401,1683,442,1851,485
%N A013945 Least d such that period of continued fraction for sqrt(d) contains n (n^2+2 if n odd, (n/2)^2+1 if n even).
%K A013945 nonn
%O A013945 1,1
%A A013945 Clark Kimberling (ck6@cedar.evansville.edu)
%I A065014
%S A065014 3,2,11,5,27,10,51,17,83,26,123,37,171,50,227,65,291,82,363,101,443,
%T A065014 122,531,145,627,170,731,197,843,226,963,257,1091,290,1227,325,1371,
%U A065014 362,1523,401,1683,442,1851,485,2027,530,2211,577,2403,626,2603,677
%N A065014 Least integer for which the periodic part of the continued fraction for its square root begins with n.
%e A065014 a(3) = 11 because the continued fraction for the square root of 11 is 3, {3, 6}.
%t A065014 a = Table[0, {70}]; Do[ b = First[ Last[ ContinuedFraction[ Sqrt[ n]]]]; If[ b < 71 && a[[b]] == 0, a[[b]] = n], {n, 2, 10^4} ]; a
%K A065014 cofr,easy,nonn
%O A065014 1,1
%A A065014 Robert G. Wilson v (rgwv@kspaint.com), Nov 01 2001
%I A052973
%S A052973 1,0,3,2,11,14,45,76,197,380,895,1838,4143,8762,19353,41496,90793,
%T A052973 195928,426811,923802,2008307,4352902,9454021,20504420,44513581,
%U A052973 96572820,209609143,454814022,987068631,2141901554,4648293425
%N A052973 A simple regular expression.
%H A052973 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1045
%F A052973 G.f.: -(-1+x)/(1-x-3*x^2+x^3)
%F A052973 Recurrence: {a(1)=0,a(0)=1,a(2)=3,a(n)-3*a(n+1)-a(n+2)+a(n+3)}
%F A052973 Sum(-1/74*(1-34*_alpha+9*_alpha^2)*_alpha^(-1-n),_alpha=RootOf(1-_Z-3*_Z^2+_Z^3))
%p A052973 spec:= [S,{S=Sequence(Prod(Union(Prod(Union(Z,Z),Sequence(Z)),Z),Z))},unlabelled ]: seq(combstruct[count ](spec,size=n),n=0..20);
%K A052973 easy,nonn
%O A052973 0,3
%A A052973 encyclopedia@pommard.inria.fr, Jan 25 2000
%E A052973 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 06 2000
%I A016560
%S A016560 3,2,12,1,4,1,3,9,13,1,1,1,1,3,1,1,1,2,2,4,52,1,1,2,1,1,
%T A016560 2,45,1,38,60,1,1,3,100,6,28,1,3,4,2,1,91,1,165,1,5,1,3,
%U A016560 1,1,1,18,2,1,22,1,5,5,3,2,1,1,1,3,2,4,1,3,2,1,39,2,1,1
%N A016560 Continued fraction for ln(65/2).
%K A016560 nonn,cofr
%O A016560 1,1
%A A016560 njas
%I A057779
%S A057779 0,0,1,0,1,1,3,2,12,14,50,98
%N A057779 Hexagonal polyominoes (or polyhexes, A000228) with perimeter 2n.
%H A057779 Andrew Clarke's Isometric Polyhexes page
%H A057779 Picture from Andrew Clarke's page showing the polyhexes of perimeters 6, 8, ... 16.
%Y A057779 Cf. A000228, A000105, A057730.
%K A057779 nonn,nice
%O A057779 1,7
%A A057779 njas, Oct 29 2000
%I A005220 M2256
%S A005220 1,0,1,0,3,2,12,14,54,86,274,528,1515,3266,8854,20422,53786,129368,
%T A005220 336103,830148,2145020
%N A005220 Dyck paths of knight moves.
%D A005220 J. Labelle and Y.-N. Yeh, Dyck paths of knight moves, Discrete Applied Math., 24 (1989), 213-221.
%K A005220 nonn,easy,nice,more
%O A005220 0,5
%A A005220 njas
%I A006774 M2257
%S A006774 1,0,3,2,12,18,65,138,432,1074,3231,8718,25999,73650,220215,643546,
%T A006774 1937877,5783700,17564727,53222094,163009086,499634508,1542392088,4770925446
%N A006774 2n-step polygons on honeycomb.
%D A006774 I. G. Enting and A. J. Guttmann, Polygons on the honeycomb lattice, J. Phys. A 22 (1989), 1371-1384.
%K A006774 nonn
%O A006774 3,3
%A A006774 njas
%I A007214 M2258
%S A007214 1,0,3,2,12,24,80,222,687,2096,6585,20892,67216,218412
%N A007214 Low temperature antiferromagnetic susceptibility for honeycomb lattice.
%D A007214 C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421.
%K A007214 look,more
%O A007214 0,3
%A A007214 sp.
%I A025232
%S A025232 3,2,12,76,504,3472,24672,179792,1337376,10117312,77618304,602528640,
%T A025232 4724294400,37361809920,297683352576,2387325283584,19255919325696,
%U A025232 156110855965696,1271401468151808
%N A025232 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3.
%F A025232 G.f.: (1-sqrt(1-12*x+28*x^2))/2 - Michael Somos, June 8, 2000.
%o A025232 (PARI) a(n)=polcoeff((1-sqrt(1-12*x+28*x^2+x*O(x^n)))/2,n)
%K A025232 nonn
%O A025232 1,1
%A A025232 Clark Kimberling (ck6@cedar.evansville.edu)
%I A055456
%S A055456 1,3,2,13,4,5,6,7,8,9,10,11,12,183,14,15,16,17,18,19,20,21,22,23,24,25,
%T A055456 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,
%U A055456 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71
%N A055456 a(n) = smallest number which is not the sum of exactly 1 or of n earlier terms.
%F A055456 If n-1>0 has not already appeared in sequence then a(n)=n-1, otherwise a(n)=n^2-n+1
%e A055456 a(3)=2 because 1 is already in the sequence, 2 has not yet appeared (i.e. is not the sum of 1 earlier term) and because the sum of 3 earlier terms is 3, 5 or 7. a(4)=13 because 1, 2 and 3 have already appeared and the sum of 4 earlier terms could be any integer from 4 through to 12.
%Y A055456 Cf. A035334. a(n) is not n-1 iff n-1 is in A024556 or equivalently in A002065.
%K A055456 easy,nonn
%O A055456 1,2
%A A055456 Henry Bottomley (se16@btinternet.com), May 19 2000
%I A005352 M2259
%S A005352 3,2,13,12,15,14,9,8,11,10,53,52,55,54,49,48,51,50,61,60,63,62,57,56,59,
%T A005352 58,37,36,39,38,33,32,35,34,45,44,47,46,41,40,43,42,213,212,215,214,209
%N A005352 Base -2 representation for -n read as binary number.
%D A005352 M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 101.
%Y A005352 Complement of A005351 in natural numbers.
%K A005352 nonn,base,nice
%O A005352 1,1
%A A005352 njas
%I A060149
%S A060149 1,3,2,13,16,106,166,1073,1934,12142
%N A060149 Number of homogeneous generators of degree n for graded algebra associated with meanders.
%H A060149 R. Bacher, Meander algebras
%Y A060149 Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.
%K A060149 nonn
%O A060149 1,2
%A A060149 njas, Apr 10 2001
%I A059374
%S A059374 1,3,2,13,18,6,73,156,108,24,501,1460,1560,720,120,4051,15030,21900,
%T A059374 15600,5400,720,37633,170142,315630,306600,163800,45360,5040,394353,
%U A059374 2107448,4763976,5891760,4292400,1834560,423360,40320
%N A059374 Triangle T(n,k)=Sum_{i=0..n} L'(n,n-i)*binomial(i,k), k=0..n-1.
%C A059374 L'(n,i) are unsigned Lah numbers (Cf. A008297).
%e A059374 [1], [3, 2], [13, 18, 6], [73, 156, 108, 24], [501, 1460, 1560, 720, 120], ... .
%Y A059374 Cf. T(n, 0) = A000262, row sums = A025168, A059110.
%K A059374 easy,nonn,tabl
%O A059374 1,2
%A A059374 Vladeta Jovovic (vladeta@Eunet.yu), Jan 28 2001
%I A064536
%S A064536 1,3,2,13,20,3,51,87,121,711,1139,3537,8034,15752,27922,49629,33201,
%T A064536 35975,143900,136341,545364,2181456,1060135,4240540,16962160,28647197,
%U A064536 13597858,205877827,100616667,381266393,1397863922,3825576990
%N A064536 Mod[Mod[4^n,3^n],2^n].
%C A064536 A generalization of A002380. It arises also as a coefficient (=c1) of 1^n=1 in a special (greedy) decomposition of 4^n into like powers as follows: 4^n=c3*3^n+c2*2^n+c1*1^n.
%F A064536 n = 7: 4^7 = 16384 = 7*[2187] + 8*[128] + 51*[1] where a(7)=51, the last coefficient; A064630[7]=7+8+a(7)=66.
%Y A064536 Cf. A002380, A064853-A064855, A060692, A064628-A064631.
%K A064536 nonn
%O A064536 1,2
%A A064536 Labos E. (labos@ana1.sote.hu), Oct 08 2001
%I A055234
%S A055234 1,3,2,14,56,6,12,42,30,168,2580,210,630,420,840,20790,416640,9240,
%T A055234 291060,83160,120120,5165160,1719277560,43825320,26860680,277560360,
%U A055234 1304863560,569729160
%N A055234 Smallest x such that sigma(x) = n*phi(x), or -1 if no such x exists.
%F A055234 a(n) = Min{x : A000203(x)/A000010(x) = n} = Min{x : A023897(x) = n}
%e A055234 sigma(14) = 24 = 4*phi(14), so a(4) = 14.
%e A055234 n = 21: a(21) = 120120 = 2.2.2.3.5.7.11.13, sigma(120120) = 483840 = n*phi(120120), phi(120120) = 23040.
%Y A055234 Cf. A000203, A000010, A020492, A023897.
%K A055234 nonn
%O A055234 1,2
%A A055234 Jud McCranie (jud.mccranie@mindspring.com), Jun 21 2000
%E A055234 a(29) > 5000000000 if it exists.
%I A033314
%S A033314 3,2,15,6,35,12,7,5,11,30,143,42,195,14,255,18,323,10,399,110,483,33,
%T A033314 23,39,27,182,87,210,899,60,1023,17,1155,34,1295,38,1443,95,1599,105,
%U A033314 1763,462,215,506,235,138,47,96,51,26,2703,78,2915,21,3135,203,3363
%N A033314 Value of D corresponding to smallest solution of Pell equation x^2-Dy^2=1 being x = 2, 3, ...
%H A033314 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A033314 Cf. A033313.
%K A033314 nonn
%O A033314 0,1
%A A033314 Eric W. Weisstein (eric@weisstein.com)
%I A051917
%S A051917 1,3,2,15,12,9,11,10,6,8,7,5,14,13,4,170,160,109,107,131,139,116,115,
%T A051917 228,234,92,8,9,73,77,220,209,85,214,80,219,199,179,203,184,66,226,70,
%U A051917 236,156,247,149,248,255,182,189,240,120,164,174,127,142,100,98,134
%N A051917 Inverse of n under Nim (or Conway) multiplication.
%C A051917 The Conway product makes N into a field of characteristic 2. This is the inverse function for that field
%D A051917 E. R. Berlekamp, J. H. Conway and R. Guy, ``Winning Ways'', p. 443
%D A051917 J. H. Conway, ``On Numbers and Games'', chapter 6
%H A051917 Index entries for sequences related to Nim-multiplication
%H A051917 Notes on game theory
%e A051917 a(4)=15 because the Conway product of 4 and 15 is 1
%K A051917 easy,nice,nonn
%O A051917 1,2
%A A051917 David A. Madore (david.madore@ens.fr), Dec 18 1999
%E A051917 More terms from John W. Layman (layman@math.vt.edu), Mar 01 2001
%I A055864
%S A055864 1,3,2,16,12,9,125,100,80,64,1296,1080,900,750,625,16807,14406,12348,
%T A055864 10584,9072,7776,262144,229376,200704,175616,153664,134456,117649,
%U A055864 4782969,4251528,3779136,3359232,2985984,2654208,2359296,2097152
%N A055864 Coefficient triangle for certain polynomials.
%C A055864 The coefficients of the partner polynomials are found in triangle A055858.
%F A055864 a(n,m)=0 if n= m >= 1;
%F A055864 E.g.f. for column m: A(m,x); A(1,x)=-(W(-x)/x+1); recursion: A(m,x) = A(m-1,x)-int(A(m-1,x),x)/x-(((m-1)^(m-1))/m)* (x^(m-1))/(m-1)!, m >= 2; W(x) principal branch of Lambert's function.
%e A055864 {1}; {3,2}; {16,12,9}; {125,100,80,64};...
%e A055864 Fourth row polynomial (n=4): p(4,x)= 125+100*x+80*x^2+64*x^3
%Y A055864 Column sequences are: A000272(n+1), n >= 1, A055865, A055070, A055867, A055868 for m=1..5.
%K A055864 easy,nonn,tabl
%O A055864 1,2
%A A055864 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Jun 20 2000
%I A026345
%S A026345 0,3,2,18,17,24,23,55,83,98,97,113,141,140,159,192,242,282,309,
%T A026345 371,420,453,507,506,542,601,600,662,727,770,769,839,885,960,
%U A026345 1038,1037,1203,1290,1289,1473,1472,1568,1631,1732,1874,1943
%N A026345 a(n) = sum of the numbers between the two n's in A026342.
%K A026345 nonn
%O A026345 1,2
%A A026345 Clark Kimberling, ck6@cedar.evansville.edu
%I A006281 M2260
%S A006281 0,3,2,18,98,33282,319994402,354455304050635218,
%T A006281 36294953231792713902640647988908098
%N A006281 Partial quotients in c.f. expansion of 2C-1, where C is Cahen's constant.
%D A006281 J. L. Davison, J. O. Shallit, Continued Fractions for Some Alternating Series, Monatsh. Math., 111 (1991), 119-126.
%Y A006281 Cf. A006279, A006280.
%K A006281 nonn
%O A006281 0,2
%A A006281 njas
%I A057026
%S A057026 3,2,19,13,17,43,103,29,67,37,41,367,199,53,463,61,131,139,73,311,163,
%T A057026 5503,89,751,97,101,211,109,113,241663,487,251,1039,2143,137,283,9343,
%U A057026 149,307,157,647,331,2719,173,1423,181,743,379,193,197,103423,823,419
%N A057026 Smallest prime of form (2n+1)*2^m-1 for some m.
%H A057026 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime
%e A057026 a(5)=43 because 2*5+1=11 and smallest prime of the form 11*2^m-1 is 43 (since 10 and 21 are not prime)
%Y A057026 Cf. A057024, A057025.
%K A057026 huge,nonn
%O A057026 0,1
%A A057026 Henry Bottomley (se16@btinternet.com), Jul 24 2000
%I A065038
%S A065038 1,3,2,20,10,41,38,75,268,247,1361,2533,3041,2751,15135,18635,51668,62443,57070,398963,181693,1313022,2359729,1034838,5365613,3225918,17353757,10212210,73599139,96446382,
%T A065038 58056874,407076917,520187758,908672243,2046244881,2712110771,6440748154,11156601694,14732275193,8416580354,41424646066,23006557538,78977395399,65854567302,107078836273,
%U A065038 188471115226,650749252297,1071511376043,872467803893,2809440878107,2402964238973,7766036476659,18849502773536,10125357598982,32332611300121,102943941995445,163227751205887,193885933162482,307443058720011,159888464280046,250759470174413,394178473635587,599819882554934,2818367283068173
%N A065038 Values of Recaman's sequence A005132 at start of n-th segment (see A064492).
%H A065038 Index entries for sequences related to Recaman's sequence
%K A065038 nonn
%O A065038 1,2
%A A065038 Allan R. Wilks (allan@research.att.com), Nov 06 2001
%I A009028
%S A009028 1,0,1,3,2,20,40,630,3092,13032,133580,1827980,21237632,244445760,
%T A009028 3154775208,45173375520,689191245168,11093076554048,189142200382672,
%U A009028 3415285850672880,65041019882791840,1301902345882783680
%V A009028 1,0,-1,3,2,-20,-40,630,-3092,13032,-133580,1827980,-21237632,244445760,
%W A009028 -3154775208,45173375520,-689191245168,11093076554048,-189142200382672,
%X A009028 3415285850672880,-65041019882791840,1301902345882783680
%N A009028 Expansion of cos(ln(1+x).cos(x)).
%t A009028 Cos[ Log[ 1+x ]*Cos[ x ] ]
%K A009028 sign,done,easy
%O A009028 0,4
%A A009028 rhh@research.bell-labs.com
%E A009028 Extended with signs 03/97 by Olivier Gerard.
%I A009022
%S A009022 1,0,1,3,2,20,74,98,1532,960,41324,105732,1595912,7998640,85401224,
%T A009022 705417112,6026865392,76352075520,537223559024,10130428275792,
%U A009022 58185728893472,1628892022801600,7352490891960224,313251680404802272
%V A009022 1,0,-1,3,-2,-20,74,98,-1532,960,41324,-105732,-1595912,7998640,85401224,
%W A009022 -705417112,-6026865392,76352075520,537223559024,-10130428275792,
%X A009022 -58185728893472,1628892022801600,7352490891960224,-313251680404802272
%N A009022 Expansion of cos(ln(1+tanh(x))).
%t A009022 Cos[ Log[ 1+Tanh[ x ] ] ]
%K A009022 sign,done,easy
%O A009022 0,4
%A A009022 rhh@research.bell-labs.com
%E A009022 Extended with signs 03/97 by Olivier Gerard.
%I A009033
%S A009033 1,0,1,3,2,20,160,1470,1412,17208,311900,5526620,27336032,64471680,
%T A009033 3978938184,91242933600,974175281648,9223539606848,209919913406800,
%U A009033 4978659648353904,83195760215366560,1324088843470088640
%V A009033 1,0,-1,3,2,-20,-160,1470,-1412,-17208,-311900,5526620,-27336032,64471680,
%W A009033 -3978938184,91242933600,-974175281648,9223539606848,-209919913406800,
%X A009033 4978659648353904,-83195760215366560,1324088843470088640
%N A009033 Expansion of cos(ln(1+x)/cosh(x)).
%t A009033 Cos[ Log[ 1+x ]/Cosh[ x ] ]
%K A009033 sign,done,easy
%O A009033 0,4
%A A009033 rhh@research.bell-labs.com
%E A009033 Extended with signs 03/97 by Olivier Gerard.
%I A018872
%S A018872 1,3,2,21,9,23,6,61,25,4,26,66,42,17,69,7,28,18,72,29,116,47,75,3,48,192,
%T A018872 77,31,124,79,5,2,32,51,129,205,13,52,33,21,53,212,134,85,34,136,86,137,
%U A018872 87,55,22,35,14,352,56,224,142,226,9,36,144,91,363,23,58,146,232,147,37
%N A018872 a(n)^5 is smallest fifth power beginning with n.
%K A018872 nonn,base
%O A018872 1,2
%A A018872 dww
%I A009574
%S A009574 0,1,1,3,2,25,129,931,7412,66753,667475,7342291,88107414,1145396473,
%T A009574 16035550517,240533257875,3848532125864,65425046139841,
%U A009574 1177650830516967,22375365779822563,447507315596451050
%V A009574 0,1,1,3,-2,25,-129,931,-7412,66753,-667475,7342291,-88107414,1145396473,
%W A009574 -16035550517,240533257875,-3848532125864,65425046139841,
%X A009574 -1177650830516967,22375365779822563,-447507315596451050
%N A009574 Expansion of sinh(ln(1+x)).exp(x).
%t A009574 Sinh[ Log[ 1+x ] ]*Exp[ x ]
%K A009574 sign,done,easy
%O A009574 0,4
%A A009574 rhh@research.bell-labs.com
%E A009574 Extended with signs 03/97 by Olivier Gerard.
%I A059422
%S A059422 1,1,0,1,1,3,2,25,213,1547,13276,129069,1375775,16009741,202184274,
%T A059422 2753591087,40231298023,627731583225,10418193719432,183264681827863,
%U A059422 3406106373633009,66695477905719251,1372395141298236250,29607108539572186329
%V A059422 1,1,0,-1,-1,3,-2,25,-213,1547,-13276,129069,-1375775,16009741,-202184274,
%W A059422 2753591087,-40231298023,627731583225,-10418193719432,183264681827863,
%X A059422 -3406106373633009,66695477905719251,-1372395141298236250,29607108539572186329
%N A059422 Difference between number of even equivalence classes and odd classes of terms in a symmetric determinant of order n.
%D A059422 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, a'_n.
%F A059422 E.g.f.: exp(1/2*t-1/4*t^2)*(1+t)^(1/2)
%Y A059422 Cf. A002135.
%K A059422 sign,done
%O A059422 0,6
%A A059422 njas, Jan 30 2001
%I A065353
%S A065353 1,0,3,2,27,90,7003,744282,14687099739,12786682083105626,
%T A065353 529158535306496354546309979,
%U A065353 7914572860144723898900437268660641289952090
%N A065353 Decimal representation of palindromes extracted from the Golden String using ever increasing Fibonacci-style subdivisions.
%C A065353 A zero must be prefixed to the 2n (n>0) terms when converting back to binary.
%H A065353 P. De Geest and J. McNamara, Palindromes in the Golden String
%H A065353 R. Knott, The Fibonacci Rabbit sequence
%e A065353 Bin (Dec) -> 1 (1); 0 (0); 11 (3); 010 (2); 11011 (27); 01011010 (90); 1101101011011 (7003); 010110101101101011010 (744282); etc.
%Y A065353 Cf. A065354, A003849, A005203, A005614, A014675, A036299.
%K A065353 nonn,huge
%O A065353 0,3
%A A065353 Patrick De Geest (pdg@worldofnumbers.com), Oct 31 2001.
%I A046272
%S A046272 0,0,3,2,29,7,5,7993,67961,769,604661,2797,78233,1306069,783641,
%T A046272 7018498457,2821109,692665944473,66841,609359,4400629,21936950640377,
%U A046272 16217038422671,3022305360281,73838133832161689,992970137,170581728179
%N A046272 Largest prime substring in 6^n (0 if none).
%Y A046272 Cf. A046264.
%K A046272 nonn
%O A046272 0,3
%A A046272 Patrick De Geest (pdg@worldofnumbers.com), Jun 1998.
%I A054676
%S A054676 1,3,2,29,67,2261,499,7601,163673,3146141,16688347,232429801,
%T A054676 1220661809,1475887019,96968880223,5041994433457,25104916552337,
%U A054676 4417388168138681,279381762131009,383174447010300497
%N A054676 Numerator of expected length of longest increasing subsequence of a permutation of length n.
%H A054676 A. M. Odlyzko and E. M. Rains, On longest increasing subsequences in random permutations, pp. 439-451 in Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis, E. L. Grinberg, S. Berhanu, M. Knopp, G. Mendoza, and E. T. Quinto, eds., Amer. Math. Soc., Contemporary Math. #251, 2000.
%Y A054676 Cf. A054677.
%K A054676 nonn,frac,nice,easy
%O A054676 1,2
%A A054676 Eric M. Rains (rains@research.att.com), Apr 19 2000
%I A062743
%S A062743 3,2,37,127,347,1087,3109,8419,24317,64553,175211,480881,1304707,
%T A062743 3523901,9558533,25874843,70115473,189961529,514272533,1394193607,
%U A062743 3779851091,10246935679,27788566133,75370121191,204475052401
%N A062743 a(n) is the prime p(m) such that Floor[p(m)/m]=n is first satisfied.
%C A062743 a(n+1)/a(n) -> e as n -> infinity, as do the m's.
%t A062743 Do[ k = 1; While[ Floor[ Prime[m]/ m] != n, m++ ]; Print[Prime[k] ], {n, 1, 27} ]
%K A062743 easy,nonn
%O A062743 0,1
%A A062743 Labos E. (labos@ana1.sote.hu), Jul 12 2001
%E A062743 More terms from Robert G. Wilson v (rgwv@kspaint.com), Jul 13 2001
%I A013324
%S A013324 1,1,0,3,2,39,12,795,2196,21086,171233,382026,13372632,50098802,
%T A013324 999044975,13582274951,35682532117,2564800895679,15018025447481,
%U A013324 416995688311441,7282021921285557,41937625525462640
%V A013324 1,1,0,-3,2,39,12,-795,-2196,21086,171233,-382026,-13372632,
%W A013324 -50098802,999044975,13582274951,-35682532117,-2564800895679,
%X A013324 -15018025447481,416995688311441,7282021921285557,-41937625525462640
%N A013324 arccosh(exp(x)-tanh(x))=x+1/2!*x^2-3/4!*x^4+2/5!*x^5+39/6!*x^6...
%K A013324 sign,done
%O A013324 0,4
%A A013324 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A009084
%S A009084 1,0,1,3,2,40,314,1134,3044,87408,733636,1835020,43592176,792796368,
%T A009084 6272579312,14019198480,1257528653744,20065121039744,124527163941136,
%U A009084 2075494412476368,75585686698926496,1110720454454795520
%V A009084 1,0,-1,3,-2,-40,314,-1134,-3044,87408,-733636,1835020,43592176,-792796368,
%W A009084 6272579312,14019198480,-1257528653744,20065121039744,-124527163941136,
%X A009084 -2075494412476368,75585686698926496,-1110720454454795520
%N A009084 Expansion of cos(tanh(ln(1+x))).
%t A009084 Cos[ Tanh[ Log[ 1+x ] ] ]
%K A009084 sign,done,easy
%O A009084 0,4
%A A009084 rhh@research.bell-labs.com
%E A009084 Extended with signs 03/97 by Olivier Gerard.
%I A065085
%S A065085 3,2,43,0,683,2731,0,43691,174763,0,2796203,44608171,0
%N A065085 Least prime having alternating bit sum (A065359) equal to -n.
%t A065085 f[n_] := (d = Reverse[ IntegerDigits[n, 2]]; l = Length[d]; s = 0; k = 1; While[k < l + 1, s = s - (-1)^k*d[[k]]; k++ ]; s); a = Table[ f[ Prime[n]], {n, 1, 10^6} ]; b = Table[0, {12} ]; Do[ If[ a[[n]] < 1 && b[[ -a[[n]] + 1]] == 0, b[[ -a[[n]] + 1]] = Prime[n]], {n, 1, 10^6} ]; b
%Y A065085 Cf. A065359.
%K A065085 base,nonn
%O A065085 0,1
%A A065085 Robert G. Wilson v (rgwv@kspaint.com), Nov 09 2001
%I A002680 M2261 N0892
%S A002680 1,3,2,45,72,105,6480,42525,22400,56133,32659200,7882875
%N A002680 Coefficients of polynomials related to Chebyshev's quadrature.
%D A002680 H. E. Salzer, Tables for facilitating the use of Chebyshev's quadrature formula, Journal of Mathematics and Physics, 26 (1947), 191-194.
%H A002680 Index entries for sequences related to Chebyshev polynomials.
%K A002680 nonn,easy,more
%O A002680 1,2
%A A002680 njas
%I A009395
%S A009395 0,1,1,3,2,45,74,1113,7288,27465,647894,1393227,59935228,604628115,
%T A009395 4418438114,143982353967,381525392432,30072901507695,405576151229266,
%U A009395 4636550476777107,187783513074129692,514355533864732725
%V A009395 0,1,1,-3,-2,45,-74,-1113,7288,27465,-647894,1393227,59935228,-604628115,
%W A009395 -4418438114,143982353967,-381525392432,-30072901507695,405576151229266,
%X A009395 4636550476777107,-187783513074129692,514355533864732725
%N A009395 Expansion of ln(1+tanh(x).exp(x)).
%t A009395 Log[ 1+Tanh[ x ]*Exp[ x ] ]
%K A009395 sign,done,easy
%O A009395 0,4
%A A009395 rhh@research.bell-labs.com
%E A009395 Extended with signs 03/97 by Olivier Gerard.
%I A063513
%S A063513 3,2,56,12,2580,630,416640,291060,1719277560
%N A063513 Least balanced numbers [A020492]: m such that the quotient Sigma[m]/Phi[m] equals the n-th prime.
%F A063513 a(n) = Min{x : A000203(x)/A000010(x) = p(n)} = Min{x : A023897(x) = p(n)}
%e A063513 n=7: p(7)=17, a(7)=416640=128.3.5.7.31 Sigma[416640]=1566720=17*Phi[a(7)], Phi[416640]=92160
%Y A063513 Smallest m such that sigma(m) = n*phi(m); (A055234) for n's which are prime.
%Y A063513 Cf. A000203, A000010, A020492, A023897, A055234.
%K A063513 nonn
%O A063513 1,1
%A A063513 Labos E. (labos@ana1.sote.hu), Jul 31 2001
%E A063513 More terms from Robert G. Wilson v (rgwv@kspaint.com), Aug 09 2001
%I A016461
%S A016461 3,2,71,12,61,1,5,12,1,5,6,6,1,9,1,1,4,1,997,1,1,1,1,8,
%T A016461 7,8,5,1,1,1,1,4,1,1,1,1,4,1,2,53,1,3,2,3,2,1,1,1,1,82,
%U A016461 1,1,2,2,6,3,260,1,1,6,1,3,24,4,1,3,6,1,4,2,1,3,1,11,2
%N A016461 Continued fraction for ln(33).
%K A016461 nonn,cofr
%O A016461 1,1
%A A016461 njas
%I A002297 M2262 N0893
%S A002297 1,1,3,2,115,11,5887,151,259723,15619,381773117,655177,20646903199,
%T A002297 27085381,467168310097,2330931341,75920439315929441,12157712239,
%U A002297 5278968781483042969,37307713155613,9093099984535515162569
%N A002297 Numerator of (2/pi)*Integral_{0..inf} (sin x / x)^n dx.
%D A002297 A. H. R. Grimsey, On the accumulation of chance effects and the Gaussian frequency distribution, Phil. Mag., 36 (1945), 294-295.
%D A002297 R. G. Medhurst and J. H. Roberts, Evaluation of the integral I_n(b) = of (2/pi)*Integral_{0..inf} (sin x / x)^n cos (bx) dx, Math. Comp., 19 (1965), 113-117.
%e A002297 1, 1, 3/4, 2/3, 115/192, 11/20, ...
%Y A002297 Cf. A002298 (for denominators), A002304, A002305. Essentially the same as A049330, except for the n=4 term.
%K A002297 nonn,frac,easy,nice
%O A002297 1,3
%A A002297 njas
%I A012860
%S A012860 3,2,756,272,958375,353792,22365525,1903757312,75760810798597,
%T A012860 29088885112832,2619341221102529187,1015423886506852352,
%U A012860 246921472831374316885,70251601603943959887872
%V A012860 -3,2,-756,272,-958375,353792,-22365525,1903757312,-75760810798597,
%W A012860 29088885112832,-2619341221102529187,1015423886506852352,
%X A012860 -246921472831374316885,70251601603943959887872
%N A012860 log(cosec(x)*tanh(x))=-3/2!*x^2+2/4!*x^4-756/6!*x^6+272/8!*x^8...
%K A012860 sign,done
%O A012860 0,1
%A A012860 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A036113
%S A036113 3,2,1312,132221,234241,24235231,1534335221,3524535231,5524634231,
%T A036113 165534534221,265544434231,265564434231,364574434221,17363574434221,
%U A036113 37262574534231,47263554635231,37365544635221,27466544634221
%N A036113 A summarize Fibonacci sequence: summarize the previous two terms!.
%C A036113 From the 76th term the sequence gets into a cycle of 3.
%Y A036113 Cf. A036059.
%K A036113 base,easy,nonn
%O A036113 0,1
%A A036113 Floor van Lamoen (f.v.lamoen@wxs.nl)
%I A027416
%S A027416 1,0,1,1,3,3
%N A027416 Rooted unlabeled trees not having a primary branch.
%D A027416 A Meir and J W Moon, On the branch-sizes of rooted unlabeled trees, in "Graph Theory and Its Applications", Annals New York Acad. Sci., Vol. 576, 1989, pp. 399-407.
%H A027416 Index entries for sequences related to rooted trees
%F A027416 The reference gives formulae.
%Y A027416 A027415 + A027416 = A000081.
%K A027416 nonn,more
%O A027416 1,5
%A A027416 njas
%I A000876
%S A000876 1,1,0,0,3,3,0,0,1,0,1,1,2,0,7,7,0,2,1,1,0,1,2,2,3,0,3,6,10,
%T A000876 0,13,2,9,3,10,0,15,15,0,10,3,9,2,13,0,10,6,3,0,3,2,2,5,2,25,
%U A000876 1,42,12,7,0,7,28,170,1,217,10,37,1
%N A000876 From a self-replicating tiling.
%Y A000876 Cf. A000360, A000361.
%K A000876 nonn
%O A000876 0,5
%A A000876 Melvyn Jeremie Lafitte (melvlafitte@hotmail.com)
%I A021307
%S A021307 0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,
%T A021307 0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,
%U A021307 3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3,3,0,0,3
%N A021307 Decimal expansion of 1/303.
%K A021307 nonn,cons
%O A021307 0,3
%A A021307 njas
%I A060523
%S A060523 1,1,0,1,1,0,3,3,0,0,9,12,3,0,0,45,60,15,0,0,0,225,345,135,15,0,0,0,
%T A060523 1575,2415,945,105,0,0,0,0,11025,18480,9030,1680,105,0,0,0,0,99225,
%U A060523 166320,81270,15120,945,0,0,0,0,0,893025,1596105,897750,217350,23625
%N A060523 Triangle T(n,k) of degree n permutations with k even cycles, k=0..n.
%F A060523 E.g.f.: (1+x)^((1-y)/2)/(1-x)^((1+y)/2).
%e A060523 [1], [1, 0], [1, 1, 0], [3, 3, 0, 0], [9, 12, 3, 0, 0], [45, 60, 15, 0, 0, 0], [225, 345, 135, 15, 0, 0, 0], [1575, 2415, 945, 105, 0, 0, 0, 0], [11025, 18480, 9030, 1680, 105, 0, 0, 0, 0], [99225, 166320, 81270, 15120, 945, 0, 0, 0, 0, 0], [893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0], ... .
%Y A060523 Cf. A060524.
%K A060523 easy,nonn,tabl
%O A060523 0,7
%A A060523 Vladeta Jovovic (vladeta@Eunet.yu), Apr 01 2001
%I A010607
%S A010607 3,3,0,1,9,2,7,2,4,8,8,9,4,6,2,6,6,8,3,8,7,4,6,0,9,9,5,2,4,0,9,0,8,
%T A010607 4,9,5,6,8,4,6,8,8,4,6,4,4,3,1,8,4,9,3,3,3,6,9,7,3,2,0,2,5,3,7,1,0,
%U A010607 9,2,7,5,6,7,5,5,7,8,8,7,1,5,3,5,8,6,5,2,5,2,6,9,5,1,6,4,8,0,9,4,5
%N A010607 Decimal expansion of cube root of 36.
%K A010607 nonn,cons
%O A010607 1,1
%A A010607 njas
%I A036477
%S A036477 3,3,0,3,3,0,3,3,3,3,3,3,3,3,3,6,6,3,6,6,6,6,6,6,9,9,6,9,9,9,1,1,1,1,1,
%T A036477 1,4,4,4,7,7,7,7,10,10,10,2,10,5,5,5,8,8,8,0,0,3,6,6,6,1,1,1,4,7,7,10,
%U A036477 2,2,5,8,0,3,6,6,1,4,4,10,2,5,8,0,3,9,1,4,10,5,5,3,6,9,4,10,2,0,3,9,7
%N A036477 partition(11n+3) mod 11.
%K A036477 nonn,part
%O A036477 1,1
%A A036477 dww
%I A021972
%S A021972 0,0,1,0,3,3,0,5,7,8,5,1,2,3,9,6,6,9,4,2,1,4,8,7,6,0,3,3,0,5,7,8,5,
%T A021972 1,2,3,9,6,6,9,4,2,1,4,8,7,6,0,3,3,0,5,7,8,5,1,2,3,9,6,6,9,4,2,1,4,
%U A021972 8,7,6,0,3,3,0,5,7,8,5,1,2,3,9,6,6,9,4,2,1,4,8,7,6,0,3,3,0,5,7,8,5
%N A021972 Decimal expansion of 1/968.
%K A021972 nonn,cons
%O A021972 0,5
%A A021972 njas
%I A019701
%S A019701 3,3,0,6,9,3,9,6,3,5,3,5,7,6,7,7,0,9,3,1,1,8,5,7,1,9,8,2,3,9,9,4,7,
%T A019701 6,7,2,0,2,0,7,5,4,6,7,3,6,1,8,4,3,2,1,9,1,6,8,1,5,7,3,1,1,4,9,7,9,
%U A019701 7,7,0,1,4,8,0,3,0,1,2,7,2,6,3,0,1,3,4,7,7,3,5,0,0,3,6,0,1,2,3,2,2
%N A019701 Decimal expansion of 2*Pi/19.
%K A019701 nonn,cons
%O A019701 0,1
%A A019701 njas
%I A031438
%S A031438 1,1,0,0,0,0,0,1,0,1,0,0,3,3,0,84,25,0
%N A031438 Nonisomorphic proper regular linear spaces, PRLIN(n).
%D A031438 A. Betten and D. Betten: The proper linear spaces on 17 points, Discrete Applied Mathematics, Volume 95, no. 1-3, 1999, pp. 83-108.
%Y A031438 Cf. A001200, A031436, A031437.
%K A031438 nonn,nice
%O A031438 0,13
%A A031438 Anton Betten (Anton.Betten@uni-bayreuth.de)
%I A051343
%S A051343 1,3,3,1,0,0,0,0,3,6,3,0,0,0,0,0,3,3,0,0,0,0,0,0,1,0,0,3,6,3,0,
%T A051343 0,0,0,0,6,6,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,
%U A051343 3,0,3,6,3,0,0,0,0,0,6,6,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,6,6
%N A051343 Ways of writing n as a sum of 3 nonnegative cubes (counted naively).
%p A051343 series(add(x^(n^3),n=0..10)^3,x,1000);
%Y A051343 Cf. A051344.
%K A051343 nonn,easy
%O A051343 0,2
%A A051343 njas
%I A059441
%S A059441 1,1,1,1,0,1,1,3,3,1,1,0,12,0,1,1,15,70,70,15,1,1,0,465,0,465,0,1,1,
%T A059441 105,3507,19355,19355,3507,105,1
%N A059441 Triangle T(n,k) (n >= 1, 0<=k<=n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.
%D A059441 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
%e A059441 1; 1,1; 1,0,1; 1,3,3,1; ...
%Y A059441 Diagonals give A001205, A002829, A005815.
%K A059441 tabl,nice,nonn
%O A059441 1,8
%A A059441 njas, Feb 01 2001
%I A059790
%S A059790 1,1,1,1,1,3,3,1,1,1,1,1,1,3,3,1,5,5,3,3,3,1,1,1,1,3,5,3,5,1,3,1,3,1,5,
%T A059790 5,1,5,3,1,1,3,1,3,3,1,1,3,3,1,1,1,3,1,5,3,3,1,3,1,3,1,1,3,5,3,1,1,3,3,
%U A059790 3,1,1,3,1,3,5,3,5,3,1,3,1,3,1,1,9,3,3,3,3,5,3,1,1,3,1,3,3,5,1,3,3,9,9
%N A059790 Distance between 2*(nth prime) and nearest prime.
%e A059790 Distance 1 means that either 2p+1 or 2p-1 is also prime.
%p A059790 with(numtheory): [seq(min(2*ithprime(k)-prevprime(2*ithprime(k)), nextprime(2*ithprime(k))-2*ithprime(k)),k=1..256)];
%Y A059790 A005382-A005385.
%K A059790 nonn
%O A059790 0,6
%A A059790 Labos E. (labos@ana1.sote.hu), Feb 22 2001
%I A016554
%S A016554 3,3,1,1,1,1,4,3,1,3,1,1,2,3,4,1,4,3,1,1,4,2,1,5,6,3,4,
%T A016554 1,7,1,1,6,2,2,19,1,4,2,13,1,3,1,51,8,1,7,3,1,64,1,1,1,
%U A016554 1,2,1,1,2,1,10,1,1,4,1,1,1,3,15,2,1,24,38,4,1,3,4,1,186
%N A016554 Continued fraction for ln(53/2).
%K A016554 nonn,cofr
%O A016554 1,1
%A A016554 njas
%I A046533
%S A046533 1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,1,1,1,1,5,1,3,5,1,1,1,1,7,11,11,7,1,1,1,
%T A046533 1,9,1,9,1,11,1,9,9,1,1,1,1,11,27,1,20,1,20,27,11,1,1,1,1,13,1,19,67,1,
%U A046533 40,67,1,19,13,1,1,1,1,15,51
%N A046533 First denominator and then numerator of 1/2-Pascal triangle (by row), excluding 2's.
%e A046533 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...
%Y A046533 Cf. A046213.
%K A046533 tabl,nonn
%O A046533 1,14
%A A046533 Mohammad K. Azarian, ma3@cedar.evansville.edu
%I A046532
%S A046532 1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,1,1,1,1,5,3,1,5,1,1,1,1,7,11,11,7,1,1,1,
%T A046532 1,9,9,1,11,1,9,1,9,1,1,1,1,11,27,20,1,20,1,27,11,1,1,1,1,13,19,1,67,
%U A046532 40,1,67,19,1,13,1,1,1,1,15,51
%N A046532 First numerator and then denominator of 1/2-Pascal triangle (by row), excluding 2's.
%e A046532 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...
%Y A046532 Cf. A046213.
%K A046532 tabl,nonn
%O A046532 1,14
%A A046532 Mohammad K. Azarian, ma3@cedar.evansville.edu
%I A014421
%S A014421 1,1,1,1,1,1,3,3,1,1,1,1,5,5,1,1,15,15,1,1,7,21,35,35,21,7,1,1,1,1,9,
%T A014421 9,1,1,45,45,1,1,11,55,165,165,55,11,1,1,495,495,1,1,13,715,1287,1287,
%U A014421 715,13,1,1,91,1001,3003,3003,1001,91,1,1,15,105,455,1365,3003,5005
%N A014421 Odd elements in Pascal's triangle.
%t A014421 Select[ Flatten[ Table[ Binomial[ n,i ],{n,0,20},{i,0,n} ] ],OddQ ]
%Y A014421 Cf. A007318, A014414.
%K A014421 nonn,easy,tabl
%O A014421 1,7
%A A014421 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A014421 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A021306
%S A021306 0,0,3,3,1,1,2,5,8,2,7,8,1,4,5,6,9,5,3,6,4,2,3,8,4,1,0,5,9,6,0,2,6,
%T A021306 4,9,0,0,6,6,2,2,5,1,6,5,5,6,2,9,1,3,9,0,7,2,8,4,7,6,8,2,1,1,9,2,0,
%U A021306 5,2,9,8,0,1,3,2,4,5,0,3,3,1,1,2,5,8,2,7,8,1,4,5,6,9,5,3,6,4,2,3,8
%N A021306 Decimal expansion of 1/302.
%K A021306 nonn,cons
%O A021306 0,3
%A A021306 njas
%I A053386
%S A053386 1,1,3,3,1,1,3,3,1,1,4,4,1,1,4,4,1,1,3,3,1,1,3,3,1,1,5,5,1,1,5,5,1,1,3,
%T A053386 3,1,1,3,3,1,1,4,4,1,1,4,4,1,1,3,3,1,1,3,3,1,1,6,6,1,1,6,6,1,1,3,3,1,1,
%U A053386 3,3,1,1,4,4,1,1,4,4,1,1,3,3,1,1,3,3,1,1,5,5,1,1,5,5,1,1,3,3,1,1,3,3,1
%N A053386 A053398(6, n).
%K A053386 nonn
%O A053386 1,3
%A A053386 dww
%I A046218
%S A046218 1,1,1,1,1,1,1,3,3,1,1,5,3,5,1,1,7,11,11,7,1,1,9,9,11,9,9,1,1,11,27,20,
%T A046218 20,27,11,1,1,13,19,67,40,67,19,13,1,1,15,51,105,147,147,105,51
%N A046218 Numerators of elements of 1/2-Pascal triangle (by row).
%e A046218 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...
%Y A046218 Cf. A046213.
%K A046218 tabl,nonn
%O A046218 1,8
%A A046218 Mohammad K. Azarian, ma3@cedar.evansville.edu
%I A046221
%S A046221 1,1,1,1,1,1,1,3,3,1,1,5,3,5,1,1,7,11,11,7,1,1,9,9,11,9,9,1,1,11,27,27,
%T A046221 11,1,1,13,19,67,67,19,13,1,1,15,51,105,147,147,105,51
%N A046221 Odd numbers on numerators of 1/2-pascal triangle (by row).
%e A046221 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...
%Y A046221 Cf. A046213.
%K A046221 tabl,nonn
%O A046221 1,8
%A A046221 Mohammad K. Azarian, ma3@cedar.evansville.edu
%I A056611
%S A056611 1,1,1,1,1,3,3,1,1,5,5,5,5,7,7,7,7,21,21,15,5,55,165,33,33,143,143,
%T A056611 1001,1001,1001,1001,91,91,221,221,221,221,323,323,323,323,2261,2261,
%U A056611 24871,24871,572033,572033,81719,81719,24035,24035,312455,312455,85215
%N A056611 Quotient: square-free kernel of A002944(n) divided by that of A001405.
%F A056611 a(n)=A007947[A002944(n)]/A007947[A001405(n)]
%e A056611 n=14,LCM[1,14,....,14,1]=24024,its sqf kernel is 6006;Binomial[14,7]=3432,its sqf kernel is 858.a(14)=6006/858=7
%Y A056611 A002944, A001405, A007947;
%K A056611 nonn
%O A056611 1,6
%A A056611 Labos E. (labos@ana1.sote.hu), Aug 07 2000
%I A034871
%S A034871 1,1,1,3,3,1,1,5,10,10,5,1,1,7,21,35,35,21,7,1,1,9,36,84,126,126,84,
%T A034871 36,9,1,1,11,55,165,330,462,462,330,165,55,11,1,1,13,78,286,715,
%U A034871 1287,1716,1716,1287,715,286,78,13,1,1,15,105,455,1365,3003,5005
%N A034871 Odd-numbered rows of Pascal's triangle.
%Y A034871 Cf. A007318.
%K A034871 nonn,tabl,easy
%O A034871 0,4
%A A034871 njas
%I A015109
%S A015109 1,1,1,1,1,1,1,3,3,1,1,5,15,5,1,1,11,55,55,11,1,1,21,231,385,231,21,1,
%T A015109 1,43,903,3311,3311,903,43,1,1,85,3655,25585,56287,25585,3655,85,1,1,
%U A015109 171,14535,208335,875007,875007,208335,14535,171,1,1,341,58311
%V A015109 1,1,1,1,-1,1,1,3,3,1,1,-5,15,-5,1,1,11,55,55,11,1,1,-21,231,-385,231,-21,1,
%W A015109 1,43,903,3311,3311,903,43,1,1,-85,3655,-25585,56287,-25585,3655,-85,1,1,
%X A015109 171,14535,208335,875007,875007,208335,14535,171,1,1,-341,58311
%N A015109 Triangle of q-binomial coefficients for q=-2.
%K A015109 sign,done,tabl,easy
%O A015109 0,8
%A A015109 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A026515
%S A026515 1,1,3,3,1,1,7,1,3,3,1,7,1,1,3,3,1,1,7,1,6,1,8,1,3,3,1,1,7,1,
%T A026515 3,3,1,7,1,1,3,3,1,8,1,6,1,7,1,1,3,3,1,1,7,1,3,3,1,7,1,1,3,3,
%U A026515 1,1,1,2,1,2,3,2,1,2,1,2,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A026515 a(n) = length of n-th run of identical symbols in A026513.
%K A026515 nonn
%O A026515 1,3
%A A026515 Clark Kimberling, ck6@cedar.evansville.edu
%I A063421
%S A063421 1,1,1,1,3,3,1,2,0,2,1,1,3,5,2,6,8,3,3,4,16,15,6,1,1,10,20,10,3,4,1,10,
%T A063421 9,15,27,15,3,4,17,60,66,32,6,1,22,41,6,71,74,36,9,1,15,6,105,168,111,
%U A063421 24,9,6,1,5,45,147,133,21
%V A063421 1,1,1,1,3,-3,1,2,0,-2,1,1,3,-5,2,6,-8,3,3,4,-16,15,-6,1,1,10,-20,10,3,-4,1,10,-9,-15,
%W A063421 27,-15,3,4,17,-60,66,-32,6,1,22,-41,-6,71,-74,36,-9,1,15,6,-105,168,-111,24,9,-6,1,5,
%X A063421 45,-147,133,21
%N A063421 Coefficient array for certain numerator polynomials N4(n,x), n >= 0 (rising powers of x) used for quadrinomials.
%C A063421 The g.f. of column k of array A008287(n,k) (quadrinomial coefficients) is (x^(ceiling(k/3)))*N4(k,x)/(1-x)^(k+1).
%C A063421 The sequence of degrees for the polynomials N4(n,x) is [0, 0, 0, 0, 2, 3, 3, 2, 5, 6, 5, 5, 8, 8, 8,...] for n >= 0.
%C A063421 Row sums N4(n,1)=1 for all n.
%F A063421 a(n,m) = [x^m]N4(n,x), n,m >= 0, with N4(n,x)= sum(((1-x)^(j-1))*(x^(b(c(n),j)))*N4(n-j,x),j=1..3), N4(n,x)= 1 for n=0,1,2, and b(c(n),j):=1 if 1<= j <= c(n) else 0, with c(n):= 2 if mod(n,3)=0 else c(n):= mod(n,3)-1; (hence b(0,j)=0, j=1..3).
%e A063421 {1};{1};{1};{1};{3,-3,1};{2,0,-2,1};{1,3,-5,2};{6,-8,3};...
%e A063421 c=1: b(1,1)=1, b(1,2)= 0 = b(1,3).
%e A063421 N4(6,x)=1+3*x-5*x^2+2*x^3.
%K A063421 sign,done,easy,tabf
%O A063421 0,5
%A A063421 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Jul 27 2001
%I A003637 M2263
%S A003637 1,1,1,1,1,3,3,1,2,1,5,1,2,3,1,7,5,3,3,1,4,5,3,4,1,5,1,5,5,3,5,7,2,5,1,
%T A003637 11,5,4,1,13,1,9,2,3,7,2,7,5,3,1,15,3,7,3,2,13,1,11,3,7,2,4,4,5,3,19,5,
%U A003637 2,6,3,9,3,5,2,19,9,4,3,17,7,4,4,1,8,3,5,9,1,21,1,15,5,7,3,5,7,7,8,25
%N A003637 Classes per genus in quadratic field with discriminant -4n+1.
%D A003637 D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
%Y A003637 Cf. A003638.
%K A003637 nonn
%O A003637 1,6
%A A003637 njas,mb
%I A004550
%S A004550 1,3,3,1,2,2,3,1,2,1,3,3,1,3,2,1,2,1,1,2,4,1,0,3,4,3,4,1,3,0,2,3,2,
%T A004550 3,3,3,3,2,0,0,3,3,4,1,2,0,0,0,4,4,4,1,2,2,1,2,2,4,4,3,1,1,4,0,0,0,
%U A004550 2,4,1,0,2,2,0,2,2,4,0,1,4,4,4,2,1,1,4,2,0,2,0,4,3,3,1,3,0,3,3,1,0
%N A004550 Expansion of sqrt(3) in base 5.
%K A004550 nonn,base,cons
%O A004550 1,2
%A A004550 njas
%I A010264
%S A010264 3,3,1,2,4,1,1,1,1,2,1,2,2,46,13,4,6,1,11,4,2,1,1,2,4,1,
%T A010264 1,11,46,1,9,8,4,1,3,52,1,13,5,1,1,1,2,3,1,1,14,1,11,11,
%U A010264 7,3,3,13,1,16,1,7,1,3,3,1,1,1,23,4,1,1,7,1,1,12,184,1
%N A010264 Continued fraction for cube root of 35.
%K A010264 nonn,cofr
%O A010264 0,1
%A A010264 njas
%I A055177
%S A055177 3,3,1,3,2,1,1,3,3,1,3,2,1,3,6,1,5,2,2,3,7,1,6,2,4,6,1,5,1,3,8,1,9,2,5,
%T A055177 6,3,5,2,7,1,4,1,3,10,1,12,2,7,6,4,5,4,7,2,4,2,8,1,9,1,3,11,1,15,2,10,
%U A055177 6,5,5,5,7,4,4,5,8,2,9,2,10,1,12,1,3,12,1
%N A055177 Cumulative counting sequence: method B (noun,adjective)-pairs with 1st term 3.
%C A055177 Segments (as in %e): 3; 3,1; 3,2,1,1; 3,3,1,3,2,1; ...
%C A055177 Conjecture: every positive integer occurs.
%e A055177 Write 3, thus having 3 1 time, thus having 3 2 times and 1 1 time, thus having 3 3 times and 1 3 times and 2 1 time, etc.
%K A055177 nonn
%O A055177 1,1
%A A055177 Clark Kimberling, ck6@cedar.evansville.edu, Apr 27 2000
%I A030778
%S A030778 3,3,1,3,2,1,3,2,1,6,5,3,2,1,7,6,5,4,3,2,1,9,8,7,6,5,4,3,2,1,
%T A030778 12,10,9,8,7,6,5,4,3,2,1,15,12,11,10,9,8,7,6,5,4,3,2,1,18,15,
%U A030778 13,12,11,10,9,8,7,6,5,4,3,2,1,21,18,16,15,13,12,11,10,9,8,7
%N A030778 Row 2, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in rows 1 and 2, for j=m,m-1,...2,1, where m=max number in row 1 from stages 1 to k-1; stage 1 is 3 in row 1.
%K A030778 nonn
%O A030778 1,1
%A A030778 Clark Kimberling, ck6@cedar.evansville.edu
%I A039992
%S A039992 1,1,1,1,1,3,3,1,3,2,3,4,1,2,2,3,2,1,2,3,4,3,2,1,3,2,4,5,2,7,6,7,11,6,
%T A039992 6,3,7,7,8,11,10,3,4,6,10,4,3,4,3,3,4,6,4,4,4,4,3,6,4,3,6,6,5,7,5,11,5,
%U A039992 7,8,4,4,7,7,7,10,3,6,10,2,1,6,4,6,3,4,3,1,5,4,4,5,6,3,6,1,4,3,4,6,3,5
%N A039992 Primes embedded in prime p(n).
%C A039992 a(n) counts permuted subsequences of digits of p(n) which denote primes.
%F A039992 a(n) = A045719(n)+1 = A039993(p(n))
%K A039992 nonn,base
%O A039992 1,6
%A A039992 Dave Wilson
%I A002332 M2264 N0894
%S A002332 0,1,3,3,1,3,5,3,7,1,9,9,5,3,9,9,3,11,1,9,11,7,15,15,13,3,15,9,11,17,
%T A002332 5,13,7,3,15,19,3,11,9,19,21,21,13,15,21,7,3,19,23,15,21,11,17,3,9,23,
%U A002332 15,13,21,25,9,5,21,23,17,27,11,25,3,19,27,27,29,9,1,5,27,17,15,21,27
%N A002332 x such that p = x^2 + 2y^2, where p = n-th prime.
%D A002332 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
%D A002332 A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
%D A002332 J. H. Jordan and J. R. Rabung, Math. Comp., 23 (1969), p. 458.
%t A002332 f[ p_ ]:=For[ y=1,True,y++,If[ IntegerQ[ x=Sqrt[ p-2y y ] ],Return[ x ] ] ]; f/@Select[ Prime/@Range[ 1,200 ],Mod[ #,8 ]<4& ]
%Y A002332 Cf. A002333.
%K A002332 nonn
%O A002332 1,3
%A A002332 njas
%E A002332 More terms from Dean Hickerson, Oct 07, 2001
%I A002102 M2265 N0895
%S A002102 1,3,3,1,3,6,3,0,3,6,6,3,1,6,6,0,3,9,6,3,6,6,3,0,3,9,12,4,0,12,6,0,3,
%T A002102 6,9,6,6,6,9,0,6,15,6,3,3,12,6,0,1,9,15,6,6,12,12,0,6,6,6,9,0,12,12,0,
%U A002102 3,18,12,3,9,12,6,0,6,9,18,7,3,12,6,0,6,15,9,9,6,12,15,0,3,21,18,6,0,6
%N A002102 Number of nonnegative solutions to x^2 + y^2 + z^2 = n.
%D A002102 H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
%D A002102 A. Das and A. C. Melissinos, Quantum Mechanics: A Modern Introduction, Gordon and Breach, 1986, p. 48.
%F A002102 Coefficient of q^k in 1/8*(1 + theta_3(0, q))^3, or coefft of q^n in (1+q+q^4+q^9+q^16+q^25+q^36+q^49+q^64+...)^3.
%t A002102 a[n_]:=Module[{x,y,z,c},For[x=c=0,x^2<=n,x++, For[y=0,x^2+y^2<=n,y++, If[IntegerQ[Sqrt[n-x^2-y^2]],c++ ]]];c]
%t A002102 CoefficientList[Series[Sum[q^n^2,{n,0,12}],{q,0,150}]^3,q]
%Y A002102 More terms from Dean Hickerson, Oct 07, 2001
%K A002102 nonn
%O A002102 0,2
%A A002102 njas
%I A047655
%S A047655 1,3,3,1,3,6,6,6,0,3,6,9,8,6,0,0,6,6,13,3,6,3,0,3,6,9,6,3,6,0,6,
%T A047655 6,3,11,0,6,0,9,0,0,0,3,13,0,0,6,0,6,3,3,6,0,15,6,3,0,6,0,6,0,6,
%U A047655 6,0,11,0,0,6,0,6,0,6,0,0,0,3,19,12,3,0,0,6,6,6,6,0,0,6,0,21,3
%V A047655 1,-3,3,-1,-3,6,-6,6,0,-3,6,-9,8,-6,0,0,-6,6,-13,3,-6,3,0,-3,6,-9,6,-3,6,0,6,
%W A047655 6,-3,11,0,6,0,9,0,0,0,-3,13,0,0,-6,0,-6,3,-3,-6,0,-15,-6,-3,0,-6,0,-6,0,-6,
%X A047655 -6,0,-11,0,0,-6,0,6,0,6,0,0,0,-3,19,12,-3,0,0,6,6,6,6,0,0,6,0,21,3
%N A047655 Expand {Product_{j=1..inf} (1-x^j) - 1 }^3 in powers of x.
%D A047655 H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
%K A047655 sign,done
%O A047655 1,2
%A A047655 njas
%I A032240
%S A032240 3,3,1,3,12,37,117,333,975,2712,7689,21414,60228,168597,475024,
%T A032240 1338525,3788400,10741575,30556305,87109332,248967446,713025093,
%U A032240 2046325125,5883406830,16944975036,48880411272,141212376513
%N A032240 Identity bracelets of n beads of 3 colors.
%H A032240 Index entries for sequences related to bracelets
%H A032240 C. G. Bower, Transforms (2)
%H A032240 F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
%F A032240 "DHK" (bracelet, identity, unlabeled) transform of 3,0,0,0...
%K A032240 nonn
%O A032240 1,1
%A A032240 Christian G. Bower (bowerc@usa.net)
%I A016604
%S A016604 3,3,1,4,1,8,6,0,0,4,6,7,2,5,2,5,6,0,9,2,4,5,4,7,0,7,8,9,7,3,3,1,4,
%T A016604 0,3,7,1,2,7,1,8,0,8,0,7,3,8,4,5,6,7,9,6,4,3,0,1,0,5,3,5,5,9,1,1,1,
%U A016604 1,3,5,8,9,8,9,6,5,1,1,9,9,7,6,2,0,9,9,3,2,5,2,5,9,0,9,9,3,9,5,7,1
%N A016604 Decimal expansion of ln(55/2).
%K A016604 nonn,cons
%O A016604 1,1
%A A016604 njas
%I A002143 M2266 N0896
%S A002143 1,1,1,1,3,3,1,5,3,1,7,5,3,5,3,5,5,3,7,1,11,5,13,9,3,7,5,15,7,13,11,3,3,
%T A002143 19,3,5,19,9,3,17,9,21,15,5,7,7,25,7,9,3,21,5,3,9,5,7,25,13,5,13,3,23,11
%N A002143 Class numbers h(-p), p = 4n-1.
%D A002143 E. T. Ordman, Tables of the class number for negative prime discriminants, Math. Comp., 23 (1969), 458.
%K A002143 nonn
%O A002143 3,5
%A A002143 njas
%I A039739
%S A039739 1,1,3,3,1,5,3,3,5,3,1,5,3,11,5,3,1,7,3,1,3,3,5,9,5,3,11,9,5,7,3,5,3,9,
%T A039739 7,1,3,11,5,15,13,3,1,5,3,3,3,27,25,21,15,13,3,5,11,5,3,1,17,15,5,7,3,
%U A039739 1,9,3,9,11,9,5,3,15,9,3,3,5,1,21,13,3,1
%N A039739 a(n)=2q-p(n), where q is the prime < p(n) for which (p(n) mod q) is maximal.
%K A039739 nonn
%O A039739 2,3
%A A039739 Clark Kimberling, ck6@cedar.evansville.edu
%I A021755
%S A021755 0,0,1,3,3,1,5,5,7,9,2,2,7,6,9,6,4,0,4,7,9,3,6,0,8,5,2,1,9,7,0,7,0,
%T A021755 5,7,2,5,6,9,9,0,6,7,9,0,9,4,5,4,0,6,1,2,5,1,6,6,4,4,4,7,4,0,3,4,6,
%U A021755 2,0,5,0,5,9,9,2,0,1,0,6,5,2,4,6,3,3,8,2,1,5,7,1,2,3,8,3,4,8,8,6,8
%N A021755 Decimal expansion of 1/751.
%K A021755 nonn,cons
%O A021755 0,4
%A A021755 njas
%I A016454
%S A016454 3,3,1,6,1,31,1,1,1,2,1461,1,1,2,1,3,1,4,2,5,1,1,104,6,
%T A016454 25,10,6,12,1,2,1,156,2,1,1,7,1,139,1,8,1,3,2,1,8,7,1,8,
%U A016454 2,3,2,9,6,46,1,3,5,1,5,1,10,1,2,7,7,2,22,2,4,3,1,177,2
%N A016454 Continued fraction for ln(26).
%K A016454 nonn,cofr
%O A016454 1,1
%A A016454 njas
%I A065227
%S A065227 1,1,3,3,1,6,2,9,2,11,21,11,23,8,22,1,17,34,7,26,46,12,34,57,15,40,66,
%T A065227 15,43,72,11,42,74,2,36,71,107,24,62,101,5,46,88,131,22,67,113,160,37,
%U A065227 86,136,187,49,102,156,1,57,114,172,231,60,121,183,246,57,122,188,255
%N A065227 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the triangular numbers. The first elements of the rows form a(n).
%Y A065227 Cf. A064766, A064865, A065221-A065226, A065228-A065234.
%K A065227 easy,nonn
%O A065227 0,3
%A A065227 Floor van Lamoen (f.v.lamoen@wxs.nl), Oct 22 2001
%I A010468
%S A010468 3,3,1,6,6,2,4,7,9,0,3,5,5,3,9,9,8,4,9,1,1,4,9,3,2,7,3,6,6,7,0,6,8,
%T A010468 6,6,8,3,9,2,7,0,8,8,5,4,5,5,8,9,3,5,3,5,9,7,0,5,8,6,8,2,1,4,6,1,1,
%U A010468 6,4,8,4,6,4,2,6,0,9,0,4,3,8,4,6,7,0,8,8,4,3,3,9,9,1,2,8,2,9,0,6,5
%N A010468 Decimal expansion of square root of 11.
%K A010468 nonn,cons
%O A010468 1,1
%A A010468 njas
%I A039798
%S A039798 1,1,1,1,3,3,1,6,14,14,1,10,40,84,84,1,15,90,300,594,594,1,21,175,
%T A039798 825,2475,4719,4719
%N A039798 Triangle of numbers of Dyck paths.
%D A039798 D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
%H A039798 Index entries for sequences related to Young tableaux.
%e A039798 1; 1,1; 1,3,3; 1,6,14,14; ...
%Y A039798 Cf. A039797.
%K A039798 nonn,tabl,easy,nice,more
%O A039798 0,5
%A A039798 njas
%I A001498
%S A001498 1,1,1,1,3,3,1,6,15,15,1,10,45,105,105,1,15,105,420,945,945,1,21,
%T A001498 210,1260,4725,10395,10395,1,28,378,3150,17325,62370,135135,135135,
%U A001498 1,36,630,6930,51975,270270,945945,2027025,2027025,1,45,990,13860
%N A001498 Triangle of coefficients of Bessel polynomials (exponents in increasing order).
%C A001498 The row polynomials with exponents in increasing order (e.g. third row: 1+3x+3x^2) are Grosswald's y_{n}(x) polynomials, p.18 eq.(7).
%D A001498 E. Grosswald, Bessel Polynomials, Lecture Notes Math. vol. 698 1978 p. 18.
%D A001498 B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
%D A001498 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A001498 Index entries for sequences related to Bessel functions or polynomials
%F A001498 a(n,0)=1; a(0,k)=0, k>0; a(n,k) = a(n-1,k)+(n-k+1)a(n,k-1) = a(n-1,k)+(n+k-1)a(n-1,k-1) [ Leonard Smiley (smiley@math.uaa.alaska.edu) ]
%F A001498 a(n,m)= A001497(n,n-m) = A001147(m)*binomial(n+m,2*m) for n >= m >= 0 else 0.
%F A001498 G.f. for m-th column: (A001147(m)*x^m)/(1-x)^(2*m+1), m >= 0, where A001147(m) = double factorials (from explicit a(n,m) form).
%p A001498 Bessel:=proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end; # explicit Bessel polynomials
%p A001498 Bessel:=proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*Bessel(n-1)+Bessel(n-2); fi; end; # recurrence for Bessel polynomials
%Y A001498 Cf. A001497, A001147, A001879, A000457, A001880, A001881. Row sums give A001515.
%K A001498 nonn,tabl,nice
%O A001498 0,5
%A A001498 njas
%I A049323
%S A049323 1,1,1,1,3,3,1,6,16,16,1,10,50,125,125,1,15,120,540,1296,1296,1,21,245,
%T A049323 1715,7203,16807,16807,1,28,448,4480,28672,114688,262144,262144,1,36,
%U A049323 756,10206,91854,551124,2125764,4782969,4782969,1,45,1200,21000,252000
%N A049323 Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.
%H A049323 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A049323 a(n,m) = A033842(n,n-m) = binomial(n+1,m+1)*(n+1)^{m-1}, n >= m >= 0, else 0.
%F A049323 p(k-1,-x)/(1-k*x)^k =(-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.
%e A049323 {1}; {1,1}; {1,3,3}; {1,6,16,16}; {1,10,50,125,125}; .... E.g. third row {1,3,3} corresponds to polynomial p{3,x)= 1+3*x+3*x^2.
%Y A049323 a(n, 0)= A000012 (powers of 1), a(n, 1)= A000217 (triangular numbers), a(n, n)= A000272(n+1), n >= 0 (diagonal); a(n, n-1)= A000272(n+1), n >= 1.
%Y A049323 For n= 0..5 the row sequences a(n, m), m >= 0, are the first columns of the triangles A023531 (unit matrix), A030528, A049324, A049325, A049326, A049327, respectively.
%Y A049323 Cf. A033842, A046757.
%K A049323 easy,nonn,tabl
%O A049323 0,5
%A A049323 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A050609
%S A050609 0,1,1,3,3,1,12,6,4,2,21,21,9,7,3,77,35,33,15,11,5,168,126,56,54,24,18,
%T A050609 8,609,273,203,91,87,39,29,13,987,987,441,329,147,141,63,47,21,3572,
%U A050609 1598,1596,714,532,238,228,102,76,34,7755,5781,2585,2583,1155,861,385
%N A050609 Sum_{i=0..x} (C(x,i) mod 2)*F(2i+y) = FL(x+y)*A050613[x], where A050613[x] = Product_{i=0..[log2(x+1)]} L(2^i)^bit(x,i).
%C A050609 Here F(n) and L(n) are n-th Fibonacci (A000045) and Lucas (A000032) numbers respectively. FL(n) is F(n) for all even n, and L(n) for all odd n.
%p A050609 generic_bincoeff_fibsum_as_sum := proc(n,k) local i; RETURN(add(((binomial(n,i) mod 2)*fibonacci(k+2*i)),i=0..n)); end;
%p A050609 generic_bincoeff_fibsum_as_product := (n,k) -> (`if`(1 = (n mod 2),luc(n+k),fibonacci(n+k)))*product('luc(2^i)^bit_i(n,i)','i'=1..floor_log_2(n+1)); # Produces same answers.
%p A050609 a(n) = generic_bincoeff_fibsum_as_sum( (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n),(n-((trinv(n)*(trinv(n)-1))/2)) ); # For trinv see A002262
%Y A050609 Transpose of A050610. First row: A051656, Second row: A050611, Third Row: A048757, Fourth row: A050612.
%K A050609 nonn,tabl
%O A050609 0,4
%A A050609 Antti.Karttunen@iki.fi 02-DEC-1999
%I A010029
%S A010029 1,1,1,3,3,1,12,11,11,56,53,3,87,321,309,53,693,2175,2119,11,680,
%T A010029 5934,17008,16687,309,8064,55674,150504,148329,53,5805,96370,572650,
%U A010029 1485465,1468457,2119,95575
%N A010029 Triangle of permutations of 1..n by number of runs of consecutive pairs up.
%D A010029 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.
%K A010029 tabl,nonn,nice
%O A010029 1,4
%A A010029 njas
%I A062746
%S A062746 1,3,3,1,12,29,30,15,3,55,222,405,417,252,84,12,273,1575,4203,6678,
%T A062746 6846,4608,1980,495,55,1428,10812,38367,83244,121518,124146,89595,
%U A062746 44990,15015,3003,273,7752,73017,325164
%V A062746 1,3,-3,1,12,-29,30,-15,3,55,-222,405,-417,252,-84,12,273,-1575,4203,-6678,6846,-4608,
%W A062746 1980,-495,55,1428,-10812,38367,-83244,121518,-124146,89595,-44990,15015,-3003,273,
%X A062746 7752,-73017,325164
%N A062746 Coefficient array for certain polynomials N(3;k,x) (rising powers of x).
%C A062746 The g.f. for the sequence of column r=2*k+1, k >= 0, of array A062745(n,r) is N(3;k,x)*(x^(k+1))/(1-x)^(2*k+2) with N(3;k,x):=sum(a(k,p)*x^p,p=0..2*k).
%C A062746 The m=0 column gives: A001764(n+1). The row sums give A000012 (powers of 1) and (unsigned) A062747.
%C A062746 The sequence of step width of this staircase array is [1,2,2,2,...], i.e. the degree of the row polynomials is [0,2,4,6,...]= A005843.
%F A062746 a(k,p):=[x^p]N(3;k,x) with N(3;k,x)=(N(3;k-1,x)-A001764(k)*(1-x)^(2*k+1))/x, N(3;0,x):=1.
%F A062746 a(n,k)= a(n-1,k+1)+((-1)^k)*binomial(2*n+1,k+1)*binomial(3*n+1,n)/(3*n+1) if k=0,..,(2*n-3); a(n,k)= ((-1)^k)*binomial(2*n+1,k+1)*binomial(3*n+1,n)/(3*n+1) if k=(2*n-2),...,2*n; else 0.
%e A062746 {1};{3,-3,1};{12,-29,30,-15,3};...; N(3;1,x)= 3-3*x+x^2.
%Y A062746 A062991.
%K A062746 sign,done,easy,tabf
%O A062746 0,2
%A A062746 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Jul 12 2001
%I A039797
%S A039797 1,1,1,3,3,1,14,14,6,1,84,84,40,10,1,594,594,300,90,15,1,4719,4719,
%T A039797 2475,825,175,21,1
%N A039797 Triangle of numbers of Dyck paths.
%D A039797 D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
%H A039797 Index entries for sequences related to Young tableaux.
%Y A039797 Cf. A039798.
%K A039797 nonn,tabl,easy,nice,more
%O A039797 0,4
%A A039797 njas
%I A001497
%S A001497 1,1,1,3,3,1,15,15,6,1,105,105,45,10,1,945,945,420,105,15,1,10395,
%T A001497 10395,4725,1260,210,21,1,135135,135135,62370,17325,3150,378,28,1,
%U A001497 2027025,2027025,945945,270270,51975,6930,630,36,1,34459425
%N A001497 Triangle of coefficients of Bessel polynomials (exponents in decreasing order).
%D A001497 E Grosswald, Bessel Polynomials, Lecture Notes Math. vol. 698 1978 p. 18.
%D A001497 B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
%D A001497 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A001497 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A001497 Index entries for sequences related to Bessel functions or polynomials
%F A001497 a(n,m)=(2*n-m)!/(m!*(n-m)!*2^(n-m)) if n >= m >= 0 else 0 (from Grosswald, p.7).
%F A001497 a(n,m)= 0, n= m >= 0 (from Grosswald p.23,(19)).
%F A001497 E.g.f. for m-th column: ((1-sqrt(1-2*x))^m)/(m!*sqrt(1-2*x)).
%p A001497 f:=proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*f(n-1)+f(n-2); fi; end;
%Y A001497 Cf. A001498. Row sums give A001515. a(n, 0)= A001147(n) (double factorials).
%K A001497 nonn,tabl,nice
%O A001497 0,4
%A A001497 njas
%I A033842
%S A033842 1,1,1,3,3,1,16,16,6,1,125,125,50,10,1,1296,1296,540,120,15,1,16807,
%T A033842 16807,7203,1715,245,21,1,262144,262144,114688,28672,4480,448,28,1,
%U A033842 4782969,4782969,2125764,551124,91854,10206,756,36,1,100000000
%N A033842 Triangle of coefficients of certain polynomials (exponents in decreasing order).
%H A033842 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A033842 a(n,m) = binomial(n+1,m)*(n+1)^(n-m-1), n >= m >= 0 else 0.
%e A033842 {1}; {1,1}; {3,3,1}; {16,16,6,1}; {125,125,50,10,1}; .... E.g. third row {3,3,1} corresponds to polynomial p{3,x)= 3*x^2+3*x+1.
%Y A033842 a(n, 0)= A000272(n+1), n >= 0 (first column); a(n, 1)= A000272(n+1), n >= 1 (second column). p(k-1, -x)/(1-k*x)^k = (-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.
%Y A033842 See also A049323.
%K A033842 easy,nonn,tabl
%O A033842 0,4
%A A033842 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A065431
%S A065431 0,3,3,1,29,4,1,20,1,2,1,45,1,1,1,5,2,1,6,1,7,9,1,12,5,4,1,22,8,1,1,1,
%T A065431 4,1,2,1,1,13,2,2,1,20,1,1,6,1,4,5,8,12,1,1,1,18,1,1,1,11,1,1813,2,1,6,
%U A065431 2,1,517,1,1,4,3,6,1,4,1,1,7,4,24,3,5,1,5,2,4,1,24,4,2,7,9,1,59,3,1,2
%N A065431 Continued fraction expansion of Hardy-Littlewood constant (A065419) product (1-(6*p^2-4*p-1)/(p-1)^4, p prime >= 5).
%H A065431 Gerhard Niklasch, Some number-theoretical constants: 1000-digit values
%t A065431 de copied from Niklasch; ContinuedFraction[de, 100]
%Y A065431 Cf. A065419.
%K A065431 nonn
%O A065431 1,2
%A A065431 Robert G. Wilson v (rgwv@kspaint.com), Nov 16 2001
%I A053375
%S A053375 1,3,3,1,39,5,273,1,4,531,7,7,12,69,5967,413,9,9,3,165,4,22419,93,
%T A053375 28,105,11,11,419775,927,6578829,1,140634693,20,105,5019135,13,
%U A053375 313191,36,123,650783,1,1153080099,4,19162705353,3,5,15,15,5,3
%N A053375 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 3 mod 4.
%C A053375 Entries are indexed by values of n from A039957.
%D A053375 R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
%Y A053375 Cf. A053370-A053375.
%K A053375 nonn,easy,nice
%O A053375 0,2
%A A053375 njas, Jan 06 2000
%I A016037
%S A016037 1,3,3,2,0,1,3,2,2,1,3,4,4,3,3,3,3,2,3,3,4,2,2,5,4,4,2,5,5,4,4,2,2,
%T A016037 5,4,4,2,5,5,4,2,3,3,3,2,2,3,3,3,2,2,3,3,3,2,2,3,3,3,2,2,3,3,3,2,2,
%U A016037 3,3,3,2,3,4,4,5,5,5,3,5,5
%N A016037 Map numbers to number of letters in English name; sequence gives number of steps before convergence.
%e A016037 1 -> 3 -> 5 -> 4, so f(1)=3.
%K A016037 nonn,word
%O A016037 0,2
%A A016037 Robert G. Wilson v (rgwv@kspaint.com)
%I A056223
%S A056223 1,1,0,1,1,1,0,1,0,1,1,1,1,1,3,3,2,1,1,0,1,1,2,2,1,0,4,4,0,1,3,
%T A056223 4,3,7,10,18,43,11,13,31,18,9,5,9,11,5,5,4,13,35,40,28,24,21,1,
%U A056223 26,38,101,42,90,33,341,241,320,199,293,87,40,48,80
%V A056223 1,1,0,-1,1,1,0,-1,0,1,-1,-1,1,-1,-3,3,2,1,-1,0,1,-1,-2,-2,1,0,-4,-4,0,1,-3,
%W A056223 -4,3,7,-10,18,43,11,-13,-31,-18,9,-5,-9,11,5,-5,-4,13,35,-40,-28,24,-21,1,
%X A056223 26,-38,-101,-42,90,33,-341,241,320,199,-293,-87,-40,48,-80
%N A056223 Hankel transform of partition numbers (A000041).
%p A056223 with(combinat): f:=(i,j) -> numbpart(i+j-2): A056223:=n->det(matrix(n,n,f));
%Y A056223 See A001906 (or Maple code here) for definition of Hankel transform. See also A051171.
%K A056223 sign,done,easy
%O A056223 1,15
%A A056223 njas, Aug 06 2000
%I A016455
%S A016455 3,3,2,1,1,1,2,2,1,4,4,1,3,10,1,4,1,5,3,5,5,1,1,1,3,108,
%T A016455 1,7,1,1,3,3,1,1,1,8,2,1,1,1,2,3,1,3,1,7,1,1,2,12,1,50,
%U A016455 4,3,23,3,8,3,1,5,1,4,7,1,6,1,1,5,1,1,1,1,6,15,1,2,5,2
%N A016455 Continued fraction for ln(27).
%K A016455 nonn,cofr
%O A016455 1,1
%A A016455 njas
(end)
Sloane's Database of Integer Sequences, Part 24
Part of the On-Line Encyclopedia of Integer Sequences
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