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{{redirect|Fibonacci Sequence|the chamber ensemble|Fibonacci Sequence (ensemble)}} | {{redirect|Fibonacci Sequence|the chamber ensemble|Fibonacci Sequence (ensemble)}} | ||
[[File:Fibonacci Squares.svg|thumb|300x300px|A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21.]] | [[File:Fibonacci Squares.svg|thumb|300x300px|A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21.]] | ||
In mathematics, the '''Fibonacci numbers''', commonly denoted {{math|''F<sub>n</sub>''}}, form a [[integer sequence|sequence]], the '''Fibonacci sequence''', in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omit the initial terms and start the sequence from 1 and 1 or from 1 and 2. | In mathematics, the '''Fibonacci numbers''', commonly denoted {{nowrap|{{math|''F<sub>n</sub>''}}{{space|hair}}}}, form a [[integer sequence|sequence]], the '''Fibonacci sequence''', in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omit the initial terms and start the sequence from 1 and 1 or from 1 and 2. | ||
Starting from 0 and 1, the next few values in the sequence are:<ref name=oeis>{{Cite OEIS|1=A000045|mode=cs2}}</ref> | Starting from 0 and 1, the next few values in the sequence are:<ref name=oeis>{{Cite OEIS|1=A000045|mode=cs2}}</ref> | ||
:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... | :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... | ||
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* At the end of the first month, they mate, but there is still only 1 pair. | * At the end of the first month, they mate, but there is still only 1 pair. | ||
* At the end of the second month they produce a new pair, so there are 2 pairs in the field.[[File:Liber abbaci magliab f124r.jpg|thumb|A page of [[Fibonacci]]'s ''[[Liber Abaci]]'' from the [[National Central Library (Florence)|Biblioteca Nazionale di Firenze]] showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.|367x367px]] | * At the end of the second month they produce a new pair, so there are 2 pairs in the field.[[File:Liber abbaci magliab f124r.jpg|thumb|A page of [[Fibonacci]]'s ''[[Liber Abaci]]'' from the [[National Central Library (Florence)|Biblioteca Nazionale di Firenze]] showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.|367x367px]] | ||
* At the end of the third month, the original pair produce a second pair, but the second pair only mate | * At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all. | ||
* At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs. | * At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs. | ||
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where | where | ||
<math display=block>\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\ldots</math> | <math display=block>\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\ldots</math> | ||
is the [[golden ratio]] | is the [[golden ratio]], and {{mvar|ψ}} is its [[Conjugate (square roots)|conjugate]]:{{Sfn | Ball | 2003 | p = 156}} | ||
<math display=block>\psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = - {1 \over \varphi} \approx -0.61803\,39887\ldots.</math> | <math display=block>\psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = - {1 \over \varphi} \approx -0.61803\,39887\ldots.</math> | ||