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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 :
or
Now, we have the next series
of equations:
or in a general case
:
By using the identity for
phi and Phi we can write:
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