Fibonacci sequence: Difference between revisions

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=== Limit of consecutive quotients ===

[[Johannes Kepler]] observed that the ratio of consecutive Fibonacci numbers [[convergent sequence|converges]]. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio <math>{{tmath|\varphi\colon </math> }}:<ref>{{Citation|last=Kepler |first=Johannes |title=A New Year Gift: On Hexagonal Snow |year=1966 |isbn=978-0-19-858120-8 |publisher=Oxford University Press |page= 92}}</ref><ref>{{Citation | title = Strena seu de Nive Sexangula | year = 1611}}</ref>

<math display=block>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>

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[[File:Fibonacci Squares.svg|frameless|260x260px]]

=== Symbolic method ===

The sequence <math>(F_n)_{n\in\mathbb N}</math> is also considered using the [[symbolic method (combinatorics)|symbolic method]].<ref>{{citation |last1=Flajolet |first1=Philippe |last2=Sedgewick |first2=Robert |title=Analytic Combinatorics|title-link= Analytic Combinatorics |date=2009 |publisher=Cambridge University Press |isbn=978-0521898065 |page=42}}</ref> More precisely, this sequence corresponds to a [[specifiable combinatorial class]]. The specification of this sequence is <math>\operatorname{Seq}(\mathcal{Z+Z^2})</math>. Indeed, as stated above, the <math>n</math>-th Fibonacci number equals the number of [[Composition (combinatorics)|combinatorial compositions]] (ordered [[integer partition|partitions]]) of <math>n-1</math> using terms 1 and 2.

It follows that the [[ordinary generating function]] of the Fibonacci sequence, <math>\sum_{i=0}^\infty F_iz^i</math>, is the [[rational function]] <math>\frac{z}{1-z-z^2}.</math>

Explicitely, if

<math display =block>S=\sum_{i=0}^\infty F_iz^i,</math>

one has

<math display =block>zS=\sum_{i=1}^\infty F_{i-1}z^i,\qquad z^2S=\sum_{i=2}^\infty F_{i-2}z^i.

</math>

So,

<math display =block>S-zS-z^2S= F_0+zF_1-zF_0=z,</math>

which is easy to solve in {{tmath|S}} to get the result.

=== Induction proofs ===

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* [[Mario Merz]] included the Fibonacci sequence in some of his artworks beginning in 1970.{{sfn|Livio|2003|p=176}}

* [[Joseph Schillinger]] (1895–1943) developed [[Schillinger System|a system of composition]] which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.{{sfn|Livio|2003|p=193}} See also {{slink|Golden ratio|Music}}.

* In [[software development]], Fibonacci numbers are often used by [[Agile management|agile]] teams operating under the [[Scrum (software development)|Scrum]] framework to size their [[product backlog]] items.<ref>{{cite web |last1=Kathuria |first1=Madhur |title=A Guide to Using the Fibonacci Sequence in Scrum |url=https://resources.scrumalliance.org/Article/guide-using-fibonacci-sequence-scrum |website=Scrum Alliance |access-date=8 August 2025}}</ref>

== See also ==