Fibonacci numbers and the Pascal Triangle


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Lucas numbers and the Golden Section



Francois-Edouard-Anatole Lucas (4.4.1842 - 8.10.1891) is the French mathematician, professor. He was educated at the Ecole Normale in Amiens. After this he worked at the Paris Observatory under Le Verrier. During the Franco-Prussian War (1870-1871) Lucas served as an artillery officer. After the French were defeated, Lucas became professor of mathematics at the Lycée Saint Louis in Paris. He later became professor of mathematics at the Lycée Charlemagne, also in Paris.


Lucas is best known for his results in number theory: in particular he studied the Fibonacci sequence and the associated Lucas sequence is named after him.Lucas is also well known for his invention of the Tower of Hanoi puzzle and other mathematical recreations. The Tower of Hanoi puzzle appeared in 1883 under the name of M. Claus. Notice that Claus is an anagram of Lucas! His four volume work on recreational mathematics Récréations mathématiques (1882-94) has become a classic. Lucas died as the result of a freak accident at a banquet when a plate was dropped and a piece flew up and cut his cheek. He died of erysipelas a few days later.

The main numerical sequence considered by Lucas is the sequence of numbers 1, 3, 4, 7, 11, 18, 29, 47, ... given with the following recurrent formula:

Ln = Ln-1 + Ln-2

for the initial terms L1 = 1 and L2 = 3.In the honor of Lucas this numerical sequence was called "Lucas numbers". Note that Lucas numbers have the same significance for mathematics, as well as the classical Fibonacci numbers.

Deviding each number in the Lucas series by the one which preceeds, we will find the following series of numbers :

3/1 = 3
4/3 = 1.3333333333
7/4 = 1.75
11/7 = 1.571428571
18/11 = 1.636363636
29/18 = 1.611111111
47/29 = 1.620689655
76/47 = 1.617021277
123//76 = 1.618421053
199/123 = 1.617886179
......... ......... .........

It also produce a ratio which stabilizes around the value of Phi :

There are interesting series of equations for Lucas numbers :

In general case we have :

We have following relations between Lucas numbers and Golden Section :

Or in general case :

There is surprising connections with Golden Section and Lucas numbers. To ilustrate this we shall now introduce operator of the finite differences that associates the function

with the function

It is easy to verify that

where D is operator of differentation.

In formula for the Lucas numbers we use the operator`s equation

which is analogue with the well known equation:

We have the identity :

As will be seen,we usually deal with next operator`s equations for the Fibonacci and Lucas numbers :


By using this equations, we can write :


Now, we have the next series of equations:

or in a general case :

By using the identity for phi and Phi we can write:





  2001-2003 Radoslav Jovanovic              translated: D.Filipovic   created:  March 2003.