Fibonacci sequence

In mathematics, the Fibonacci numbers or Fibonacci sequence is a well-known sequence of integers with various interesting properties.

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Description

The Fibonacci series is made up of the following integer sequence:

<math>1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\;</math>

or (often, in modern usage):

<math>0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\;</math> Template:OEIS2C.

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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_n of Fibonacci numbers is defined by the recurrence relation

<math>F_n = F_{n-1} + F_{n-2},\!\,</math>

with seed valuesTemplate:SfnTemplate:Sfn

<math>F_1 = 1,\; F_2 = 1</math>

orTemplate:Sfn

<math>F_0 = 0,\; F_1 = 1.</math>

The Fibonacci sequence is named after Italian mathematician Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics,Template:Sfn although the sequence had been described earlier in Indian mathematics.<ref name="GlobalScience" /><ref name = "HistoriaMathematica" /><ref name="Donald Knuth 2006 50" /> By modern convention, the sequence begins either with F₀ = 0 or with F₁ = 1. The Liber Abaci began the sequence with F₁ = 1.

Fibonacci numbers are closely related to Lucas numbers <math>L_n</math> in that they form a complementary pair of Lucas sequences <math>U_n(1,-1)=F_n</math> and <math>V_n(1,-1)=L_n</math>. 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, ... . 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,<ref name="S. Douady and Y. Couder 1996 255–274">Template:Citation</ref> such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple,<ref name="Jones 2006 544">Template:Citation</ref> the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.<ref name = "A. Brousseau 1969 525–532">Template:Citation</ref>

See also

• Mathematics
• Pattern
• Spiral

External links

• Fibonacci number @ Wikipedia