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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 ''F<sub>n</sub>'' 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 

''F''<sub>0</sub>&nbsp;=&nbsp;0 or with ''F''<sub>1</sub>&nbsp;=&nbsp;1.

The ''Liber Abaci'' began the sequence with ''F''<sub>1</sub>&nbsp;=&nbsp;1.

== Lucas numbers ==

Fibonacci numbers are closely related to [[Lucas number]]s L_n in that they form a complementary pair of [[Lucas sequence]]s 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 approximation]]s to the ratio are 2/1,&nbsp;3/2,&nbsp;5/3,&nbsp;8/5,&nbsp;...&nbsp;.

== Applications ==

Applications include computer algorithms such as the [[Fibonacci search technique]] and the [[Fibonacci heap]] data structure, and graphs called [[Fibonacci cube]]s 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 ==

* [[Mathematical beauty]]
* [[Mathematics]]
* [[Mathematics and art]]
* [[Pascal's triangle]]
* [[Pattern]]
* [[Spiral]]

== External links == 

* [https://en.wikipedia.org/wiki/Fibonacci_number Fibonacci number] @ Wikipedia

[[Category:Fibonacci sequence]]
[[Category:Mathematical beauty]]
[[Category:Mathematics]]