FranklynDavitt2: slightly fixed citations for clarity also and updated info to improve readability

2024-05-07T16:44:03Z

slightly fixed citations for clarity also and updated info to improve readability

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2022-06-29T23:53:39Z

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← Older revision		Revision as of 05:23, 30 June 2022
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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 ~~without breeding~~, so there are 3 pairs in all.		* 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]] ~~({{OEIS2C\|id=A001622}})~~, ~~while~~ {{mvar\|ψ}} is its [[~~Conjugate_~~(~~square_roots~~)\|conjugate]], which both display [[Golden_ratio_conjugate\|the unique properties of the golden ratio]], as −{{mvar\|ψ}} is also the [[Multiplicative_inverse\|inverse]] of {{mvar\|φ}} and its decimal part over 1 at the same time:{{Sfn \| Ball \| 2003 \| p = 156}}		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>

Fibonacci number - Revision history

FranklynDavitt2: slightly fixed citations for clarity also and updated info to improve readability

69.127.115.14: robot: Update article (please report if you notice any mistake or error in this edit)

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