Fibonacci sequence
In mathematics, the Fibonacci numbers or Fibonacci sequence is a well-known sequence of integers with various interesting properties.
Description
By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation
- F_n = F_{n-1} + F_{n-2},\!\,
with seed values
- F_1 = 1,\; F_2 = 1
or
- F_0 = 0,\; F_1 = 1.
History
The Fibonacci sequence is named after Italian mathematician Fibonacci.
His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.
By modern convention, the sequence begins either with
F0 = 0 or with F1 = 1.
The Liber Abaci began the sequence with F1 = 1.
Lucas numbers
Fibonacci numbers are closely related to Lucas numbers L_n in that they form a complementary pair of Lucas sequences U_n(1,-1)=F_n and V_n(1,-1)=L_n.
They are intimately connected with the golden ratio.
For example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... .
Applications
Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.
They also appear in biological settings, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.
See also
External links
- Fibonacci number @ Wikipedia
