In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted F_n . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins

The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.

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, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone's bracts, though they do not occur in all species.

Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.

The Fibonacci numbers may be defined by the recurrence relation $${\displaystyle F_{0}=0,\quad F_{1}=1,}$$ and $${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$$ for n > 1.

Under some older definitions, the value ${\displaystyle F_{0}=0}$ is omitted, so that the sequence starts with ${\displaystyle F_{1}=F_{2}=1,}$ and the recurrence ${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$ is valid for n > 2.

The first 20 Fibonacci numbers F_n are:

The Fibonacci sequence can be extended to negative integer indices by following the same recurrence relation in the negative direction (sequence A039834 in the OEIS): ⁠${\displaystyle F_{1}=1}$⁠, ⁠${\displaystyle F_{0}=0}$⁠, and ⁠${\displaystyle F_{n}=F_{n+2}-F_{n+1}}$⁠ for n < 0 . Nearly all properties of Fibonacci numbers do not depend upon whether the indices are positive or negative. The values for positive and negative indices obey the relation: $${\displaystyle F_{-n}=(-1)^{n+1}F_{n}.}$$