In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:

0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; (sequence A000045 in OEIS)

By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s.

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_0 = 0 \quad\text{and}\quad F_1 = 1.

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonacci, "son of Bonaccio"). Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence may have been previously described in Indian mathematics.

Fibonacci numbers are used in the analysis of financial markets, in strategies such as Fibonacci retracement, and are used in computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure. The simple recursion of Fibonacci numbers has also inspired a family of recursive graphs called Fibonacci cubes for interconnecting parallel and distributed systems. They also appear in biological settings,[2] such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[3] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone