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

- (sequence A000045 in OEIS).

By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

In mathematical terms, the sequence *F*_{n} of Fibonacci numbers is defined by the recurrence relation

with seed values^{[1]}

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci’s 1202 book *Liber Abaci* introduced the sequence to Western European mathematics,^{[2]} although the sequence had been described earlier in Indian mathematics.^{[3]}^{[4]}^{[5]} (By modern convention, the sequence begins with *F*_{0}Â =Â 0. The *Liber Abaci* began the sequence with *F*_{1}Â =Â 1, omitting the initial 0, and the sequence is still written this way by some.)

Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair of Lucas sequences. They are intimately connected with the golden ratio, for example the closest rational approximations to the ratio are 2/1,Â 3/2,Â 5/3,Â 8/5,Â …Â . 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,^{[6]} such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit spouts of a pineapple,^{[7]} the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.^{[8]}

A tiling with squares whose sides are successive Fibonacci numbers in length

A Fibonacci spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34. See golden spiral.

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