Wikipedia
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Often, especially in modern usage, the sequence is extended by one more initial term:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
.
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 F of Fibonacci numbers is defined by the recurrence relation
F = F + F,
with seed values
F = 1, F = 1
or
F = 0, F = 1.
The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier as Virahanka numbers in Indian mathematics. By modern convention, the sequence begins either with F = 0 or with F = 1. The sequence described in Liber Abaci began with F = 1.
Fibonacci numbers are closely related to Lucas numbers L in that they form a complementary pair of Lucas sequences U(1, − 1) = F and V(1, − 1) = L. 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, ... .
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers 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, such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.