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Fibonacci

Sunday, May 16th, 2010

Fibonacci Number

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

$0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots.$

By definition, the first two Fibonacci numbers are 0 and 1, and each remaining 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 F_n 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 Bonaccio, “son of Bonaccio”). Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.

List of Fibonacci numbers

The first 21 Fibonacci numbers, also denoted as F_n, for n = 0, 1, 2, … ,20 are:

F₀	F₁	F₂	F₃	F₄	F₅	F₆	F₇	F₈	F₉	F₁₀	F₁₁	F₁₂	F₁₃	F₁₄	F₁₅	F₁₆	F₁₇	F₁₈	F₁₉	F₂₀
0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	610	987	1597	2584	4181	6765

Using the recurrence relation, the sequence can also be extended to negative index n. The result satisfies the equation

$F_{-n} = (-1)^{n+1} F_n. \!\,$

Thus the complete sequence is

$\ldots,\;-8,\;5,\;-3,\;2,\;-1,\;1,\;0,\;1,\;1,\;2,\;3,\;5,\;8,\;\ldots$

Source: Wikipedia