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{{Short description|Numbers obtained by adding the two previous ones}}
{{For|the chamber ensemble|Fibonacci Sequence (ensemble)}}
{{pp-pc}}

In mathematics, the '''Fibonacci sequence''' is a [[Integer sequence|sequence]] in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as '''Fibonacci numbers''', commonly denoted {{nowrap|{{math|''F<sub>n</sub>''}}{{space|hair}}}}. Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1<ref>Richard A. Brualdi, ''Introductory Combinatorics'', Fifth edition, Pearson, 2005</ref><ref>Peter Cameron, ''Combinatorics: Topics, Techniques, Algorithms'', Cambridge University Press, 1994</ref> and some (as did [[Fibonacci]]) from 1 and 2. Starting from 0 and 1, the sequence begins
: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... {{OEIS|A000045}}

[[File:Fibonacci Squares.svg|thumb|A tiling with [[square]]s whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21]]
The Fibonacci numbers were first described in [[Indian mathematics]] as early as 200&nbsp;BC in work by [[Pingala]] on enumerating possible patterns of [[Sanskrit]] poetry formed from syllables of two lengths.<ref name="GlobalScience" /><ref name="HistoriaMathematica" /><ref name="Donald Knuth 2006 50" /> They are named after the Italian mathematician Leonardo of Pisa, also known as [[Fibonacci]], who introduced the sequence to Western European mathematics in his 1202 book {{lang|la|[[Liber Abaci]]}}.{{Sfn|Sigler|2002|pp=404–05}}

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''[[Fibonacci Quarterly]]''. Applications of Fibonacci numbers include computer algorithms such as the [[Fibonacci search technique]] and the [[Fibonacci heap]] [[data structure]], and [[graph (discrete mathematics)|graphs]] called [[Fibonacci cube]]s used for interconnecting parallel and distributed systems. They also appear [[Patterns in nature#Spirals|in biological settings]], such as branching in trees, [[phyllotaxis|the arrangement of leaves on a stem]], the fruit sprouts of a [[pineapple]], the flowering of an [[artichoke]], and the arrangement of a [[pine cone]]'s bracts, though they do not occur in all species.

Fibonacci numbers are also strongly related to the [[golden ratio]]: [[#Binet's formula|Binet's formula]] expresses the {{mvar|n}}-th Fibonacci number in terms of {{mvar|n}} and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as {{mvar|n}} increases. Fibonacci numbers are also closely related to [[Lucas number]]s, which obey the same [[recurrence relation]] and with the Fibonacci numbers form a complementary pair of [[Lucas sequence]]s.

==Definition==
[[File:Fibonacci Spiral.svg|thumb|The Fibonacci spiral: an approximation of the [[golden spiral]] created by drawing [[circular arc]]s connecting the opposite corners of squares in the Fibonacci tiling (see preceding image)]]

The Fibonacci numbers may be defined by the [[recurrence relation]]{{Sfn | Lucas | 1891 | p=3}}
<math display=block>F_0=0,\quad F_1= 1,</math>
and
<math display=block>F_n=F_{n-1} + F_{n-2}</math>
for {{math|''n'' > 1}}.

Under some older definitions, the value <math>F_0 = 0</math> is omitted, so that the sequence starts with <math>F_1=F_2=1,</math> and the recurrence <math>F_n=F_{n-1} + F_{n-2}</math> is valid for {{math|''n'' > 2}}.{{Sfn | Beck | Geoghegan | 2010}}{{Sfn | Bóna | 2011 | p=180}} <!--Fibonacci started the sequence with index 0: {{math|<sub>0</sub>→1, <sub>1</sub>→2, <sub>2</sub>→3, ..., <sub>12</sub>→377}}.<ref>{{citation |last1=Leonardo da Pisa |title=File:Liber abbaci magliab f124r.jpg - Wikimedia Commons |date=1202 |url=https://commons.wikimedia.org/wiki/File:Liber_abbaci_magliab_f124r.jpg |language=en}}</ref>-->

The first 20 Fibonacci numbers {{math|''F<sub>n</sub>''}} are:
:{| class="wikitable" style="text-align:right"
! {{math|''F''<sub>0</sub>}}
! {{math|''F''<sub>1</sub>}}
! {{math|''F''<sub>2</sub>}}
! {{math|''F''<sub>3</sub>}}
! {{math|''F''<sub>4</sub>}}
! {{math|''F''<sub>5</sub>}}
! {{math|''F''<sub>6</sub>}}
! {{math|''F''<sub>7</sub>}}
! {{math|''F''<sub>8</sub>}}
! {{math|''F''<sub>9</sub>}}
! {{math|''F''<sub>10</sub>}}
! {{math|''F''<sub>11</sub>}}
! {{math|''F''<sub>12</sub>}}
! {{math|''F''<sub>13</sub>}}
! {{math|''F''<sub>14</sub>}}
! {{math|''F''<sub>15</sub>}}
! {{math|''F''<sub>16</sub>}}
! {{math|''F''<sub>17</sub>}}
! {{math|''F''<sub>18</sub>}}
! {{math|''F''<sub>19</sub>}}
|-
| 0
| 1
| 1
| 2
| 3
| 5
| 8
| 13
| 21
| 34
| 55
| 89
| 144
| 233
| 377
| 610
| 987
| 1597
| 2584
| 4181
|}

The Fibonacci sequence can be extended to negative integer indices by following the same recurrence relation in the negative direction {{OEIS|A039834}}: {{tmath|1= F_1=1}}, {{tmath|1= F_0= 0}}, and {{tmath|1= F_{n}=F_{n+2} - F_{n+1} }} for {{math|''n'' < 0 }}. Nearly all properties of Fibonacci numbers do not depend upon whether the indices are positive or negative. The values for positive and negative indices obey the relation:<ref>{{cite book
| last = Vajda | first = Steven | author-link = Steven Vajda
| year = 1989
| title = Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications
| place = Chichester | publisher = Ellis Horwood 
| isbn = 0-7458-0715-1
| page = 10
}}</ref>
<math display=block> F_{-n} = (-1)^{n+1} F_n. </math>

== History ==

===India===
{{see also|Golden ratio#History}}
[[File:Fibonacci Sanskrit prosody.svg|thumb|Thirteen ({{math|''F''<sub>7</sub>}}) ways of arranging long and short syllables in a cadence of length six. Eight ({{math|''F''<sub>6</sub>}}) end with a short syllable and five ({{math|''F''<sub>5</sub>}}) end with a long syllable.]]

The Fibonacci sequence appears in [[Indian mathematics]], in connection with [[Sanskrit prosody]].<ref name="HistoriaMathematica">{{Citation|first=Parmanand|last=Singh|title= The So-called Fibonacci numbers in ancient and medieval India|journal=Historia Mathematica|volume=12|issue=3|pages=229–244|year=1985|doi = 10.1016/0315-0860(85)90021-7|doi-access=free}}</ref><ref name="knuth-v1">{{Citation|title=The Art of Computer Programming|volume=1|first=Donald|last=Knuth| author-link =Donald Knuth |publisher=Addison Wesley|year=1968|isbn=978-81-7758-754-8|url=https://books.google.com/books?id=MooMkK6ERuYC&pg=PA100|page=100|quote=Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns&nbsp;... both Gopala (before 1135&nbsp;AD) and Hemachandra (c.&nbsp;1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)&nbsp;...}}</ref>{{sfn|Livio|2003|p=197}} In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration {{mvar|m}} units is {{math|''F''<sub>''m''+1</sub>}}.<ref name="Donald Knuth 2006 50">{{Citation|title = The Art of Computer Programming | volume = 4. Generating All Trees – History of Combinatorial Generation | first = Donald | last = Knuth | author-link = Donald Knuth |publisher= Addison–Wesley |year= 2006 | isbn= 978-0-321-33570-8 | page = 50 | url= https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50 | quote = it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats.  ... there are exactly Fm+1 of them.  For example the 21 sequences when {{math|1=''m'' = 7}} are: [gives list].  In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)}}</ref>

Knowledge of the Fibonacci sequence was expressed as early as [[Pingala]] ({{circa}}&nbsp;450&nbsp;BC–200&nbsp;BC). Singh cites Pingala's cryptic formula ''misrau cha'' ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for {{mvar|m}} beats ({{math|''F''<sub>''m''+1</sub>}}) is obtained by adding one [S] to the {{math|''F''<sub>''m''</sub>}} cases and one [L] to the {{math|''F''<sub>''m''−1</sub>}} cases.<ref>{{Citation | last = Agrawala | first = VS | year = 1969 | title = ''Pāṇinikālīna Bhāratavarṣa'' (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan | quote = SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala  [1969,  463–76],  after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC}}</ref> [[Bharata Muni]] also expresses knowledge of the sequence in the ''[[Natya Shastra]]'' ({{Circa}}&nbsp;100&nbsp;BC–{{Circa}}&nbsp;350&nbsp;AD).<ref name=GlobalScience>{{Citation|title=Toward a Global Science|first=Susantha|last=Goonatilake|author-link=Susantha Goonatilake|publisher=Indiana University Press|year=1998|page=126|isbn=978-0-253-33388-9|url=https://books.google.com/books?id=SI5ip95BbgEC&pg=PA126}}</ref><ref name="HistoriaMathematica"/>
However, the clearest exposition of the sequence arises in the work of [[Virahanka]] ({{Circa}}&nbsp;700&nbsp;AD), whose own work is lost, but is available in a quotation by Gopala ({{Circa}}&nbsp;1135):{{sfn|Livio|2003|p=197}}

<blockquote>Variations of two earlier meters [is the variation]&nbsp;...  For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]&nbsp;... In this way, the process should be followed in all ''mātrā-vṛttas'' [prosodic combinations].{{efn|"For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier—three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven [[Mora (linguistics)|morae]] [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" <ref>{{Citation|last=Velankar|first=HD|year=1962|title='Vṛttajātisamuccaya' of kavi Virahanka|publisher=Rajasthan Oriental Research Institute|location=Jodhpur|page=101}}</ref>}}</blockquote>

[[Hemachandra]] ({{Circa}}&nbsp;1150) is credited with knowledge of the sequence as well,<ref name=GlobalScience/> writing that "the sum of the last and the one before the last is the number&nbsp;... of the next mātrā-vṛtta."{{sfn|Livio|2003|p=197–198}}<ref>{{citation|last1=Shah|first1=Jayant|year=1991|title=A History of Piṅgala's Combinatorics|url=https://web.northeastern.edu/shah/papers/Pingala.pdf|publisher=[[Northeastern University]]|page=41|access-date=2019-01-04}}</ref>

===Europe===
[[File:Liber abbaci magliab f124r.jpg|thumb|upright=1.25|A page of [[Fibonacci]]'s {{lang|la|[[Liber Abaci]]}} from the [[National Central Library (Florence)|Biblioteca Nazionale di Firenze]] showing (in box on right) 13 entries of the Fibonacci sequence:<br /> the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.]]

The Fibonacci sequence first appears in the book {{lang|la|[[Liber Abaci]]}} (''The Book of Calculation'', 1202) by [[Fibonacci]],{{Sfn|Sigler|2002|pp=404–405}}<ref>{{citation|url=https://www.math.utah.edu/~beebe/software/java/fibonacci/liber-abaci.html|title=Fibonacci's Liber Abaci (Book of Calculation)|date=13 December 2009|website=[[The University of Utah]]|access-date=2018-11-28}}</ref> where it is used to calculate the growth of rabbit populations.<ref>{{citation
 | last = Tassone | first = Ann Dominic
 | date = April 1967
 | doi = 10.5951/at.14.4.0285
 | issue = 4
 | journal = The Arithmetic Teacher
 | jstor = 41187298
 | pages = 285–288
 | title = A pair of rabbits and a mathematician
 | volume = 14}}</ref> Fibonacci considers the growth of an idealized ([[biology|biologically]] unrealistic) [[rabbit]] population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the rabbit [[Mathematical problem|math problem]]: how many pairs will there be in one year?

* 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.
* 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 {{mvar|n}}-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month {{math|''n'' – 2}}) plus the number of pairs alive last month (month {{math|''n'' – 1}}). The number in the {{mvar|n}}-th month is the {{mvar|n}}-th Fibonacci number.<ref>{{citation | last = Knott | first = Ron
 | title = Fibonacci's Rabbits | url=http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits | publisher =[[University of Surrey]] Faculty of Engineering and Physical Sciences}}</ref>

The name "Fibonacci sequence" was first used by the 19th-century number theorist [[Édouard Lucas]].<ref>{{Citation | first = Martin | last = Gardner | author-link = Martin Gardner |title=Mathematical Circus |publisher = The Mathematical Association of America |year=1996 |isbn= 978-0-88385-506-5 | quote = It is ironic that Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Édouard Lucas...  attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber abaci | page = 153}}</ref>

[[File:Fibonacci Rabbits.svg|left|thumb|upright=1.5|Solution to Fibonacci rabbit [[Mathematical problem|problem]]: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At ''the end of the n''th month, the number of pairs is equal to ''F<sub>n.</sub>'']]
{{clear|left}}

== Relation to the golden ratio ==
{{also|Golden ratio|Golden field}}

===Closed-form expression <span class="anchor" id="Binet's formula"></span>===
Like every [[sequence]] defined by a homogeneous [[linear recurrence with constant coefficients]], the Fibonacci numbers have a [[closed-form expression]].<ref>{{cite book |title=Discrete Mathematics with Ducks |first=sarah-marie|last=belcastro|author-link=Sarah-Marie Belcastro |edition=2nd |publisher=CRC Press |year=2018 |isbn=978-1-351-68369-2 |page=260 |url=https://books.google.com/books?id=xoqADwAAQBAJ}} [https://books.google.com/books?id=xoqADwAAQBAJ&pg=PA260 Extract of page 260]</ref> It has become known as '''Binet's formula''', named after French mathematician [[Jacques Philippe Marie Binet]], though it was already known by [[Abraham de Moivre]] and [[Daniel Bernoulli]]:<ref>{{citation | last1 = Beutelspacher | first1 = Albrecht | last2 = Petri | first2 = Bernhard | contribution = Fibonacci-Zahlen | doi = 10.1007/978-3-322-85165-9_6 | pages = 87–98 | publisher = Vieweg+Teubner Verlag | title = Der Goldene Schnitt | series = Einblick in die Wissenschaft | year = 1996| isbn = 978-3-8154-2511-4 }}</ref>

<math display=block>
F_n = \frac{\varphi^n-\psi^n}{\varphi-\psi} = \frac{\varphi^n-\psi^n}{\sqrt 5},
</math>

where {{tmath|\varphi}} is the [[golden ratio]] and {{tmath|\psi}} is its [[Conjugate (square roots)|conjugate]],{{Sfn | Ball | 2003 | p = 156}}

<math display=block>\begin{align}
\varphi &= \tfrac12\bigl(1 + \sqrt{5}~\!\bigr)= \phantom{-}1.61803\ldots, \\[5mu]
\psi &= \tfrac12\bigl(1 - \sqrt{5}~\!\bigr) = -0.61803\ldots.
\end{align}</math>
The numbers {{tmath|\varphi}} and {{tmath|\psi}} are the two solutions of the [[quadratic equation]] {{tmath|1=\textstyle x^2 - x - 1 = 0}}, that is, {{tmath|1= (x - \varphi)(x - \psi) = x^2 - x - 1}}, and thus they satisfy the identities {{tmath|1= \varphi + \psi = 1}} and {{tmath|1= \varphi\psi = -1}}.

Since <math>\psi = -\varphi^{-1}</math>, Binet's formula can also be written as

<math display=block>
F_n = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt 5} = \frac{\varphi^n - (-\varphi)^{-n}}{2\varphi - 1}.
</math>

To see the relation between the sequence and these constants,{{Sfn | Ball | 2003 | pp = 155–156}} note that <math>\varphi</math> and <math>\psi</math> are also roots of <math>x^n = x^{n-1} + x^{n-2},</math> so the powers of <math>\varphi</math> and <math>\psi</math> satisfy the Fibonacci recurrence. In other words,

<math display=block>\begin{align}
\varphi^n &= \varphi^{n-1} + \varphi^{n-2}, \\[3mu]
\psi^n &= \psi^{n-1} + \psi^{n-2}.
\end{align}</math>

It follows that for any values {{mvar|a}} and {{mvar|b}}, the sequence defined by

<math display=block>U_n=a \varphi^n + b \psi^n</math>

satisfies the same recurrence. If {{mvar|a}} and {{mvar|b}} are chosen so that {{math|1=''U''<sub>0</sub> = 0}} and {{math|1=''U''<sub>1</sub> = 1}} then the resulting sequence {{math|''U''<sub>''n''</sub>}} must be the Fibonacci sequence. This is the same as requiring {{mvar|a}} and {{mvar|b}} satisfy the system of equations:

<math display=block>
\begin{align} a \varphi^0 + b \psi^0 &= 0 \\  a \varphi^1 + b \psi^1 &= 1\end{align}
</math>

which has solution

<math display=block>
a = \frac{1}{\varphi-\psi} = \frac{1}{\sqrt 5},\quad  b = -a,
</math>

producing the required formula.

Taking the starting values {{math|''U''<sub>0</sub>}} and {{math|''U''<sub>1</sub>}} to be arbitrary constants and solving the system of equations gives the general solution
<math display=block>\begin{align}
a&=\frac{U_1-U_0\psi}{\sqrt 5}, \\[3mu]
b&=\frac{U_0\varphi-U_1}{\sqrt 5}.
\end{align}</math>
In particular, choosing {{math|1=''a'' = 1}} makes the {{mvar|n}}-th element of the sequence closely approximate the {{mvar|n}}-th power of {{tmath|\varphi}} for large enough values of {{mvar|n}}. This arises when {{math|1=''U''{{sub|0}} = 2}} and {{math|1=''U''{{sub|1}} = 1}}, which produces the sequence of [[Lucas number]]s.

=== Computation by rounding ===
Since
<math display=inline>\left|\frac{\psi^{n}}{\sqrt 5}\right| < \frac{1}{2}</math> for all {{math|''n'' ≥ 0}}, the number {{math|''F''<sub>''n''</sub>}} is the closest [[integer]] to <math>\frac{\varphi^n}{\sqrt 5}</math>. Therefore, it can be found by [[rounding]], using the nearest integer function:
<math display=block>F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5}\right\rceil,\ n \geq 0.</math>

In fact, the rounding error quickly becomes very small as {{mvar|n}} grows, being less than 0.1 for {{math|''n'' ≥ 4}}, and less than 0.01 for {{math|''n'' ≥ 8}}. This formula is easily inverted to find an index of a Fibonacci number {{mvar|F}}:
<math display=block>n(F) = \left\lfloor  \log_\varphi \sqrt{5}F\right\rceil,\ F \geq 1.</math>

Instead using the [[floor function]] gives the largest index of a Fibonacci number that is not greater than {{mvar|F}}:
<math display=block>n_{\mathrm{largest}}(F) = \left\lfloor  \log_\varphi \sqrt{5}(F+1/2)\right\rfloor,\ F \geq 0,</math>
where <math>\log_\varphi(x) = \ln(x)/\ln(\varphi) = \log_{10}(x)/\log_{10}(\varphi)</math>, <math>\ln(\varphi) = 0.481211\ldots</math>,<ref>{{Cite OEIS|1=A002390|2=Decimal expansion of natural logarithm of golden ratio|mode=cs2}}</ref> and <math>\log_{10}(\varphi) = 0.208987\ldots</math>.<ref>{{Cite OEIS|1=A097348|2=Decimal expansion of arccsch(2)/log(10)|mode=cs2}}</ref>

=== Magnitude ===
Since ''F<sub>n</sub>'' is [[Asymptotic analysis|asymptotic]] to <math>\varphi^n/\sqrt5</math>, the number of digits in {{math|''F''<sub>''n''</sub>}} is asymptotic to <math>n\log_{10}\varphi\approx 0.2090\, n</math>. As a consequence, for every integer {{math|''d'' > 1}} there are either 4 or 5 Fibonacci numbers with {{mvar|d}} decimal digits.

More generally, in the [[radix|base]] {{mvar|b}} representation, the number of digits in {{math|''F''<sub>''n''</sub>}} is asymptotic to <math>n\log_b\varphi = \frac{n \log \varphi}{\log b}.</math>

=== 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 {{tmath|\varphi}}:<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>

This convergence holds regardless of the starting values <math>U_0</math> and <math>U_1</math>, unless <math>U_1 = -U_0/\varphi</math>. This can be verified using [[#Binet's formula|Binet's formula]]. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ...&thinsp;. The ratio of consecutive elements in this sequence shows the same convergence towards the golden ratio.

In general, <math>\lim_{n\to\infty}\frac{F_{n+m}}{F_n}=\varphi^m
</math>, because the ratios between consecutive Fibonacci numbers approaches <math>\varphi</math>.

: [[File:Fibonacci tiling of the plane and approximation to Golden Ratio.gif|thumb|upright=2.2|left|Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous]]
{{Clear}}

=== Decomposition of powers ===
Since the golden ratio satisfies the equation
<math display=block>\varphi^2 = \varphi + 1,</math>

this expression can be used to decompose higher powers <math>\varphi^n</math> as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of <math>\varphi</math> and 1. The resulting [[recurrence relation]]ships yield Fibonacci numbers as the linear [[coefficient]]s:
<math display=block>\varphi^n = F_n\varphi + F_{n-1}.</math>
This equation can be [[Mathematical proof|proved]] by [[Mathematical induction|induction]] on {{math|''n'' ≥ 1}}:
<math display=block>\begin{align}
\varphi^{n+1} &= (F_n\varphi + F_{n-1})\varphi = F_n\varphi^2 + F_{n-1}\varphi \\
&= F_n(\varphi+1) + F_{n-1}\varphi = (F_n + F_{n-1})\varphi + F_n = F_{n+1}\varphi + F_n.
\end{align}</math>
For <math>\psi = -1/\varphi</math>, it is also the case that <math>\psi^2 = \psi + 1</math> and it is also the case that
<math display=block>\psi^n = F_n\psi + F_{n-1}.</math>

These expressions are also true for {{math|''n'' < 1}} if the Fibonacci sequence ''F<sub>n</sub>'' is [[Generalizations of Fibonacci numbers#Extension to negative integers|extended to negative integers]] using the Fibonacci rule <math>F_n = F_{n+2} - F_{n+1}.</math>

=== Identification ===
Binet's formula provides a proof that a positive integer {{mvar|x}} is a Fibonacci number [[if and only if]] at least one of <math>5x^2+4</math> or <math>5x^2-4</math> is a [[Square number|perfect square]].<ref>{{Citation | title = Fibonacci is a Square | last1 = Gessel | first1 = Ira | journal = [[The Fibonacci Quarterly]] | volume = 10 | issue = 4 | pages = 417–19 |date=October 1972 | url = https://www.fq.math.ca/Scanned/10-4/advanced10-4.pdf | access-date = 2012-04-11 }}</ref> This is because Binet's formula, which can be written as <math>F_n = (\varphi^n - (-1)^n \varphi^{-n}) / \sqrt{5}</math>, can be multiplied by <math>\sqrt{5} \varphi^n</math> and solved as a [[quadratic equation]] in <math>\varphi^n</math> via the [[quadratic formula]]:

<math display=block>\varphi^n = \frac{F_n\sqrt{5} \pm \sqrt{5{F_n}^{\!2} + 4(-1)^n}}{2}.</math>

Comparing this to <math>\varphi^n = F_n \varphi + F_{n-1} = (F_n\sqrt{5} + F_n + 2 F_{n-1})/2</math>, it follows that
: <math display=block>5{F_n}^{\!2} + 4(-1)^n = (F_n + 2F_{n-1})^2\,.</math>
In particular, the left-hand side is a perfect square.

== Matrix form ==
A 2-dimensional system of linear [[difference equation]]s that describes the Fibonacci sequence is

<math display=block>
\begin{pmatrix} F_{k+2} \\ F_{k+1} \end{pmatrix}
= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} F_{k+1} \\ F_{k}\end {pmatrix} </math>
alternatively denoted
<math display=block> \vec F_{k+1} = \mathbf{A} \vec F_{k},</math>

which yields <math>\vec F_n = \mathbf{A}^n  \vec F_0</math>. The [[eigenvalue]]s of the [[matrix (mathematics)|matrix]] {{math|'''A'''}} are <math>\varphi=\tfrac12\bigl(1+\sqrt5~\!\bigr)</math> and <math>\psi=-\varphi^{-1}=\tfrac12\bigl(1-\sqrt5~\!\bigr)</math> corresponding to the respective [[eigenvector]]s
<math display=block>\vec \mu=\begin{pmatrix} \varphi \\ 1 \end{pmatrix}, \quad \vec\nu=\begin{pmatrix} -\varphi^{-1} \\ 1 \end{pmatrix}.</math>

As the initial value is
<math display=block>
  \vec F_0=\begin{pmatrix} 1 \\ 0 \end{pmatrix}=\frac{1}{\sqrt{5}}\vec{\mu}\,-\,\frac{1}{\sqrt{5}}\vec{\nu},
</math>
it follows that the {{mvar|n}}th element is
<math display=block>\begin{align}
  \vec F_n\ &= \frac{1}{\sqrt{5}}A^n\vec\mu-\frac{1}{\sqrt{5}}A^n\vec\nu \\
  &= \frac{1}{\sqrt{5}}\varphi^n\vec\mu - \frac{1}{\sqrt{5}}(-\varphi)^{-n}\vec\nu \\
  &= \cfrac{1}{\sqrt{5}}\left(\cfrac{1+\sqrt{5}}{2}\right)^{\!n}\begin{pmatrix} \varphi \\ 1 \end{pmatrix} \,-\, \cfrac{1}{\sqrt{5}}\left(\cfrac{1-\sqrt{5}}{2}\right)^{\!n}\begin{pmatrix}{c} -\varphi^{-1} \\ 1 \end{pmatrix}.
\end{align}</math>

From this, the {{mvar|n}}th element in the Fibonacci series may be read off directly as a [[closed-form expression]]:
<math display=block>
F_n = \cfrac{1}{\sqrt{5}}\left(\cfrac{1+\sqrt{5}}{2}\right)^{\!n} - \, \cfrac{1}{\sqrt{5}}\left(\cfrac{1-\sqrt{5}}{2}\right)^{\!n}.
</math>

Equivalently, the same computation may be performed by [[Matrix diagonalization|diagonalization]] of {{math|'''A'''}} through use of its [[eigendecomposition]]:
<math display=block>\begin{align} 
  A & = S\Lambda S^{-1}, \\[3mu]
  A^n & = S\Lambda^n S^{-1},
\end{align}</math>
where
<math display=block>
  \Lambda=\begin{pmatrix} \varphi & 0 \\ 0 & -\varphi^{-1}\! \end{pmatrix}, \quad
  S=\begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}.
</math>
The closed-form expression for the {{mvar|n}}th element in the Fibonacci series is therefore given by
<math display=block>\begin{align} 
  \begin{pmatrix} F_{n+1} \\ F_n \end{pmatrix} & = 
  A^{n} \begin{pmatrix} F_1 \\ F_0 \end{pmatrix}\ 
  \\ & = S \Lambda^n S^{-1} \begin{pmatrix} F_1 \\ F_0 \end{pmatrix} 
  \\ & = S \begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix} S^{-1} 
    \begin{pmatrix} F_1 \\ F_0 \end{pmatrix} 
  \\ & = \begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}
     \begin{pmatrix}\varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix}
     \frac{1}{\sqrt{5}}\begin{pmatrix} 1 & \varphi^{-1} \\ -1 & \varphi \end{pmatrix} 
     \begin{pmatrix} 1 \\ 0 \end{pmatrix},
\end{align}</math>
which again yields
<math display=block>
  F_n = \cfrac{\varphi^n-(-\varphi)^{-n}}{\sqrt{5}}.
</math>

The matrix {{math|'''A'''}} has a [[determinant]] of −1, and thus it is a 2&thinsp;×&thinsp;2 [[unimodular matrix]].

This property can be understood in terms of the [[continued fraction]] representation for the golden ratio {{mvar|φ}}:
<math display=block>
  \varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}.
</math>
The [[convergent (continued fraction)|convergents]] of the continued fraction for {{mvar|φ}} are ratios of successive Fibonacci numbers: {{math|1=''φ''<sub>''n''</sub> = ''F''<sub>''n''+1</sub> / ''F''<sub>''n''</sub>}} is the {{mvar|n}}-th convergent, and the {{math|(''n''&thinsp;+&thinsp;1)}}-st convergent can be found from the recurrence relation {{math|1=''φ''<sub>''n''+1</sub> = 1 + 1 / ''φ''<sub>''n''</sub>}}.<ref>{{Cite web |title=The Golden Ratio, Fibonacci Numbers and Continued Fractions. |url=https://nrich.maths.org/2737 |access-date=2024-03-22 |website=nrich.maths.org |language=en}}</ref> The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:
<math display=block>
  \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = 
  \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}.
</math>
For a given {{mvar|n}}, this matrix can be computed in {{math|''O''(log ''n'')}} arithmetic operations,{{efn|This counts [[arbitrary-precision arithmetic]] operations as being {{math|''O''(1)}}. If taking the bitlength into account, exponentiation by squaring is still a marked improvement, but overall complexity is dominated by the very last multiplication step; there are {{math|''O''(''n'')}} digits in the result, and the task requires producing all of them.}} using the [[exponentiation by squaring]] method.

Taking the determinant of both sides of this equation yields [[Cassini's identity]],
<math display=block>(-1)^n = F_{n+1}F_{n-1} - {F_n}^2.</math>

Moreover, since {{math|'''A'''<sup>''n''</sup>'''A'''<sup>''m''</sup> {{=}} '''A'''<sup>''n''+''m''</sup>}} for any [[square matrix]] {{math|'''A'''}}, the following [[identity (mathematics)|identities]] can be derived (they are obtained from two different coefficients of the [[matrix product]], and one may easily deduce the second one from the first one by changing {{mvar|n}} into {{math|''n'' + 1}}),
<math display=block>\begin{align}
 {F_m}{F_n} + {F_{m-1}}{F_{n-1}} &= F_{m+n-1}, \\[3mu]
 F_{m} F_{n+1} + F_{m-1} F_n &= F_{m+n} .
\end{align}</math>

In particular, with {{math|1=''m'' = ''n''}},
<math display=block>\begin{align}
 F_{2 n-1} &= {F_n}^2 + {F_{n-1}}^2 \\[6mu]
 F_{2 n\phantom{{}-1}}   &= (F_{n-1}+F_{n+1})F_n \\[3mu]
          &= (2 F_{n-1}+F_n)F_n \\[3mu]
          &= (2 F_{n+1}-F_n)F_n.
\end{align}</math>

These last two identities provide a way to compute Fibonacci numbers [[Recursion (computer science)|recursively]] in {{math|''O''(log ''n'')}} arithmetic operations. This matches the time for computing the {{mvar|n}}-th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with [[memoization]]).<ref>{{citation|title=In honour of Fibonacci|first=Edsger W.|last=Dijkstra|author-link=Edsger W. Dijkstra|year=1978|url=https://www.cs.utexas.edu/users/EWD/ewd06xx/EWD654.PDF}}</ref>

== Combinatorial identities ==

=== Combinatorial proofs ===
Most identities involving Fibonacci numbers can be proved using [[combinatorial proof|combinatorial arguments]] using the fact that <math>F_n</math> can be interpreted as the number of (possibly empty) sequences of&nbsp;1s and&nbsp;2s whose sum is <math>n-1</math>. This can be taken as the definition of <math>F_n</math> with the conventions <math>F_0 = 0</math>, meaning no such sequence exists whose sum is&nbsp;−1, and <math>F_1 = 1</math>, meaning the empty sequence "adds up" to 0. In the following, <math>|{...}|</math> is the [[cardinality]] of a [[set (mathematics)|set]]:

: <math>F_0 = 0 = |\{\}|</math>
: <math>F_1 = 1 = |\{()\}|</math>
: <math>F_2 = 1 = |\{(1)\}|</math>
: <math>F_3 = 2 = |\{(1,1),(2)\}|</math>
: <math>F_4 = 3 = |\{(1,1,1),(1,2),(2,1)\}|</math>
: <math>F_5 = 5 = |\{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\}|</math>

In this manner the recurrence relation
<math display=block>F_n = F_{n-1} + F_{n-2}</math>
may be understood by dividing the <math>F_n</math> sequences into two non-overlapping sets where all sequences either begin with 1 or 2:
<math display=block>F_n = |\{(1,...),(1,...),...\}| + |\{(2,...),(2,...),...\}|</math>
Excluding the first element, the remaining terms in each sequence sum to <math>n-2</math> or <math>n-3</math> and the cardinality of each set is <math>F_{n-1}</math> or <math>F_{n-2}</math> giving a total of <math>F_{n-1}+F_{n-2}</math> sequences, showing this is equal to <math>F_n</math>.

In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the {{mvar|n}}-th is equal to the {{math|(''n'' + 2)}}-th Fibonacci number minus&nbsp;1.{{Sfn | Lucas | 1891 | p = 4}}   In symbols:
<math display=block>\sum_{i=1}^n F_i = F_{n+2} - 1</math>

This may be seen by dividing all sequences summing to <math>n+1</math> based on the location of the first 2. Specifically, each set consists of those sequences that start <math>(2,...), (1,2,...), ..., </math> until the last two sets <math>\{(1,1,...,1,2)\}, \{(1,1,...,1)\}</math> each with cardinality 1.

Following the same logic as before, by summing the cardinality of each set we see that
: <math>F_{n+2} = F_n + F_{n-1} + ... + |\{(1,1,...,1,2)\}| + |\{(1,1,...,1)\}|</math>
... where the last two terms have the value <math>F_1 = 1</math>. From this it follows that <math>\sum_{i=1}^n F_i = F_{n+2}-1</math>.

A similar argument, grouping the sums by the position of the first&nbsp;1 rather than the first&nbsp;2 gives two more identities:
<math display=block>\sum_{i=0}^{n-1} F_{2 i+1} = F_{2 n}</math>
and
<math display=block>\sum_{i=1}^{n} F_{2 i} = F_{2 n+1}-1.</math>
In words, the sum of the first Fibonacci numbers with [[parity (mathematics)|odd]] index up to <math>F_{2 n-1}</math> is the {{math|(2''n'')}}-th Fibonacci number, and the sum of the first Fibonacci numbers with [[parity (mathematics)|even]] index up to <math>F_{2 n}</math> is the {{math|(2''n'' + 1)}}-th Fibonacci number minus&nbsp;1.<ref>{{Citation|title = Fibonacci Numbers |last1 = Vorobiev |first1 = Nikolaĭ Nikolaevich |first2 = Mircea|last2= Martin |publisher = Birkhäuser |year = 2002 |isbn = 978-3-7643-6135-8 |chapter=Chapter 1 |pages = 5–6}}</ref>

A different trick may be used to prove
<math display=block>\sum_{i=1}^n F_i^2 = F_n F_{n+1}</math>
or in words, the sum of the squares of the first Fibonacci numbers up to <math>F_n</math> is the product of the {{mvar|n}}-th and {{math|(''n'' + 1)}}-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size <math>F_n \times F_{n+1}</math> and decompose it into squares of size <math>F_n, F_{n-1}, ..., F_1</math>; from this the identity follows by comparing areas:

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

=== Induction proofs ===
Fibonacci identities often can be easily proved using [[mathematical induction]].

For example, reconsider
<math display=block>\sum_{i=1}^n F_i = F_{n+2} - 1.</math>
Adding <math>F_{n+1}</math> to both sides gives
: <math>\sum_{i=1}^n F_i + F_{n+1} = F_{n+1} + F_{n+2} - 1</math>
and so we have the formula for <math>n+1</math>
<math display=block>\sum_{i=1}^{n+1} F_i = F_{n+3} - 1</math>

Similarly, add <math>{F_{n+1}}^2</math> to both sides of
<math display=block>\sum_{i=1}^n F_i^2 = F_n F_{n+1}</math>
to give
<math display=block>\sum_{i=1}^n F_i^2 + {F_{n+1}}^2 = F_{n+1}\left(F_n + F_{n+1}\right)</math>
<math display=block>\sum_{i=1}^{n+1} F_i^2 = F_{n+1}F_{n+2}</math>

=== Binet formula proofs ===

The Binet formula is
<math display=block>\sqrt5F_n = \varphi^n - \psi^n.</math>
This can be used to prove Fibonacci identities.

For example, to prove that <math display=inline>\sum_{i=1}^n F_i = F_{n+2} - 1</math>
note that the left hand side multiplied by <math>\sqrt5</math> becomes
<math display=block>
\begin{align}
1 +& \varphi + \varphi^2 + \dots + \varphi^n - \left(1 + \psi + \psi^2 + \dots + \psi^n \right)\\
&= \frac{\varphi^{n+1}-1}{\varphi-1} - \frac{\psi^{n+1}-1}{\psi-1}\\
&= \frac{\varphi^{n+1}-1}{-\psi} - \frac{\psi^{n+1}-1}{-\varphi}\\
&= \frac{-\varphi^{n+2}+\varphi + \psi^{n+2}-\psi}{\varphi\psi}\\
&= \varphi^{n+2}-\psi^{n+2}-(\varphi-\psi)\\
&= \sqrt5(F_{n+2}-1)\\
\end{align}</math>
as required, using the facts <math display=inline>\varphi\psi =- 1</math> and <math display=inline>\varphi-\psi=\sqrt5</math> to simplify the equations.

== Other identities ==
Numerous other identities can be derived using various methods. Here are some of them:<ref name="MathWorld">{{MathWorld|urlname=FibonacciNumber |title=Fibonacci Number|mode=cs2}}</ref>

=== Cassini's and Catalan's identities ===
{{Main|Cassini and Catalan identities}}

Cassini's identity states that
<math display=block>{F_n}^2 - F_{n+1}F_{n-1} = (-1)^{n-1}</math>
Catalan's identity is a generalization:
<math display=block>{F_n}^2 - F_{n+r}F_{n-r} = (-1)^{n-r}{F_r}^2</math>

=== d'Ocagne's identity ===
<math display=block>F_m F_{n+1} - F_{m+1} F_n = (-1)^n F_{m-n}</math>
<math display=block>F_{2 n} = {F_{n+1}}^2 - {F_{n-1}}^2 = F_n \left (F_{n+1}+F_{n-1} \right ) = F_nL_n</math>
where {{math|''L''<sub>''n''</sub>}} is the {{mvar|n}}-th [[Lucas number]]. The last is an identity for doubling {{mvar|n}}; other identities of this type are
<math display=block>F_{3 n} = 2{F_n}^3 + 3 F_n F_{n+1} F_{n-1} = 5{F_n}^3 + 3 (-1)^n F_n</math>
by Cassini's identity.

<math display=block>F_{3 n+1} = {F_{n+1}}^3 + 3 F_{n+1}{F_n}^2 - {F_n}^3</math>
<math display=block>F_{3 n+2} = {F_{n+1}}^3 + 3 {F_{n+1}}^2 F_n + {F_n}^3</math>
<math display=block>F_{4 n} = 4 F_n F_{n+1} \left ({F_{n+1}}^2 + 2{F_n}^2 \right ) - 3{F_n}^2 \left ({F_n}^2 + 2{F_{n+1}}^2 \right )</math>
These can be found experimentally using [[lattice reduction]], and are useful in setting up the [[special number field sieve]] to [[Factorization|factorize]] a Fibonacci number.

More generally,<ref name="MathWorld" />

<math display=block>F_{k n+c} = \sum_{i=0}^k \binom k i F_{c-i} {F_n}^i {F_{n+1}}^{k-i}.</math>

or alternatively

<math display=block>F_{k n+c} = \sum_{i=0}^k \binom k i F_{c+i} {F_n}^i {F_{n-1}}^{k-i}.</math>

Putting {{math|1=''k'' = 2}} in this formula, one gets again the formulas of the end of above section [[#Matrix form|Matrix form]].

== Generating function ==
The [[ordinary generating function]] of the Fibonacci sequence is the [[power series]]

<math display=block>
s(z) = \sum_{k=0}^\infty F_k z^k = 0 + z + z^2 + 2z^3 + 3z^4 + 5z^5 + \cdots.
</math>

This series is convergent for any [[complex number]] <math>z</math> satisfying <math>|z| < 1/\varphi \approx 0.618,</math> and its sum has a simple closed form:<ref>{{Citation | last = Glaister | first = P | title = Fibonacci power series | journal = The Mathematical Gazette | year = 1995 | doi = 10.2307/3618079 | volume = 79 | issue = 486| pages = 521–25 | jstor = 3618079 | s2cid = 116536130 }}</ref>

<math display=block>s(z)=\frac{z}{1-z-z^2}.</math>

This can be proved by multiplying by <math display="inline">(1-z-z^2)</math>:
<math display=block>\begin{align}
(1 - z- z^2) s(z)
  &= \sum_{k=0}^{\infty} F_k z^k - \sum_{k=0}^{\infty} F_k z^{k+1} - \sum_{k=0}^{\infty} F_k z^{k+2} \\
  &= \sum_{k=0}^{\infty} F_k z^k - \sum_{k=1}^{\infty} F_{k-1} z^k - \sum_{k=2}^{\infty} F_{k-2} z^k \\
  &= 0z^0 + 1z^1 - 0z^1 + \sum_{k=2}^{\infty} (F_k - F_{k-1} - F_{k-2}) z^k \\
  &= z,
\end{align}</math>

where all terms involving <math>z^k</math> for <math>k \ge 2</math> cancel out because of the defining Fibonacci recurrence relation.

The [[partial fraction decomposition]] is given by
<math display=block>s(z) = \frac{1}{\sqrt5}\left(\frac{1}{1 - \varphi z} - \frac{1}{1 - \psi z}\right)</math>
where <math display=inline>\varphi = \tfrac12\left(1 + \sqrt{5}\right)</math> is the golden ratio and <math>\psi = \tfrac12\left(1 - \sqrt{5}\right)</math> is its [[Conjugate (square roots)|conjugate]].

Using <math>z</math> equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of <math>s(z)</math>. For example, <math>s(0.001) = \frac{0.001}{0.998999} = \frac{1000}{998999} = 0.001001002003005008013021\ldots.</math>

== Reciprocal sums ==
<!--
Borwein credits some formulae to {{Citation | author = Landau, E. | title = Sur la Série des Invers de Nombres de Fibonacci | journal = Bull. Soc. Math. France | volume = 27 | year = 1899 | pages = 298–300}}
-->
Infinite sums over [[multiplicative inverse|reciprocal]] Fibonacci numbers can sometimes be evaluated in terms of [[theta function]]s. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as
<math display=block>\sum_{k=1}^\infty \frac{1}{F_{2 k-1}} = \frac{\sqrt{5}}{4} \; \vartheta_2\!\left(0, \frac{3-\sqrt 5}{2}\right)^2 ,</math>

and the sum of squared reciprocal Fibonacci numbers as
<math display=block>\sum_{k=1}^\infty \frac{1}{{F_k}^2} = \frac{5}{24} \!\left(\vartheta_2\!\left(0, \frac{3-\sqrt 5}{2}\right)^4 - \vartheta_4\!\left(0, \frac{3-\sqrt 5}{2}\right)^4 + 1 \right).</math>

If we add 1 to each Fibonacci number in the first sum, there is also the closed form
<math display=block>\sum_{k=1}^\infty \frac{1}{1+F_{2 k-1}} = \frac{\sqrt{5}}{2},</math>

and there is a ''nested'' sum of squared Fibonacci numbers giving the reciprocal of the [[golden ratio]],
<math display=block>\sum_{k=1}^\infty \frac{(-1)^{k+1}}{\sum_{j=1}^k {F_{j}}^2} = \frac{\sqrt{5}-1}{2} .</math>

The sum of all even-indexed reciprocal Fibonacci numbers is<ref>[[Edmund Landau|Landau]] (1899)<!-- most probably: {{Citation | author = Landau, E. | title = Sur la Série des Invers de Nombres de Fibonacci | journal = Bull. Soc. Math. France | volume = 27 | year = 1899 | pages = 298–300}}
--> quoted according [[#Borwein|Borwein]], Page 95, Exercise 3b.</ref>
<math display=block>\sum_{k=1}^{\infty} \frac{1}{F_{2 k}} = \sqrt{5} \left(L(\psi^2) - L(\psi^4)\right) </math>
with the [[Lambert series]] <math>\textstyle L(q) := \sum_{k=1}^{\infty} \frac{q^k}{1-q^k} ,</math> since <math>\textstyle \frac{1}{F_{2 k}} = \sqrt{5} \left(\frac{\psi^{2 k}}{1-\psi^{2 k}} - \frac{\psi^{4 k}}{1-\psi^{4 k}} \right)\!.</math>

So the [[reciprocal Fibonacci constant]] is<ref>{{Cite OEIS|1=A079586|2=Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the {{mvar|k}}-th Fibonacci number|mode=cs2}}</ref>
<math display=block>\sum_{k=1}^{\infty} \frac{1}{F_k} = \sum_{k=1}^\infty \frac{1}{F_{2 k-1}} + \sum_{k=1}^{\infty} \frac {1}{F_{2 k}} = 3.359885666243 \dots</math>

Moreover, this number has been proved [[irrational number|irrational]] by [[Richard André-Jeannin]].<ref>{{citation
 | last = André-Jeannin
 | first = Richard
 | title = Irrationalité de la somme des inverses de certaines suites récurrentes
 | journal = [[Comptes Rendus de l'Académie des Sciences, Série I]]
 | volume = 308
 | year = 1989
 | issue = 19
 | pages = 539–41
 |mr=0999451}}</ref>

'''Millin's series''' gives the identity<ref>{{citation|title=Mathematical Gems III|volume=9|series=Dolciani Mathematical Expositions|first=Ross|last=Honsberger|publisher=American Mathematical Society|year=1985|isbn=9781470457181|contribution=Millin's series|pages=135–136|contribution-url=https://books.google.com/books?id=vl_0DwAAQBAJ&pg=PA135}}</ref>
<math display=block>\sum_{k=0}^{\infty} \frac{1}{F_{2^k}} = \frac{7 - \sqrt{5}}{2},</math>
which follows from the closed form for its partial sums as {{mvar|N}} tends to infinity:
<math display=block>\sum_{k=0}^N \frac{1}{F_{2^k}} = 3 - \frac{F_{2^N-1}}{F_{2^N}}.</math>

== Primes and divisibility ==

=== Divisibility properties ===
Every third number of the sequence is even (a multiple of <math>F_3=2</math>) and, more generally, every {{mvar|k}}-th number of the sequence is a multiple of ''F<sub>k</sub>''. Thus the Fibonacci sequence is an example of a [[divisibility sequence]]. In fact, the Fibonacci sequence satisfies the stronger divisibility property<ref>{{Citation | first = Paulo | last = Ribenboim | author-link = Paulo Ribenboim | title = My Numbers, My Friends | publisher = Springer-Verlag | year = 2000}}</ref><ref>{{Citation | last1 = Su | first1 = Francis E | others = et al | publisher = HMC | url = http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml | contribution = Fibonacci GCD's, please | year = 2000 | title = Mudd Math Fun Facts | access-date = 2007-02-23 | archive-url = https://web.archive.org/web/20091214092739/http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml | archive-date = 2009-12-14 | url-status = dead }}</ref>
<math display=block>\gcd(F_a,F_b,F_c,\ldots) = F_{\gcd(a,b,c,\ldots)}\,</math>
where {{math|gcd}} is the [[greatest common divisor]] function. (This relation is different if a different indexing convention is used, such as the one that starts the sequence with {{tmath|1=F_0 = 1}} and {{tmath|1=F_1 = 1}}.)

In particular, any three consecutive Fibonacci numbers are pairwise [[Coprime integers|coprime]] because both <math>F_1=1</math> and <math>F_2 = 1</math>. That is,
: <math>\gcd(F_n, F_{n+1}) = \gcd(F_n, F_{n+2}) = \gcd(F_{n+1}, F_{n+2}) = 1</math>
for every {{mvar|n}}.

Every [[prime number]] {{mvar|p}} divides a Fibonacci number that can be determined by the value of {{mvar|p}} [[modular arithmetic|modulo]]&nbsp;5. If {{mvar|p}} is congruent to 1 or 4 modulo 5, then {{mvar|p}} divides {{math|''F''<sub>''p''−1</sub>}}, and if {{mvar|p}} is congruent to 2 or 3 modulo 5, then, {{mvar|p}} divides {{math|''F''<sub>''p''+1</sub>}}. The remaining case is that {{math|1=''p'' = 5}}, and in this case {{mvar|p}} divides ''F<sub>p</sub>''.

<math display=block>\begin{cases} p =5 & \Rightarrow p \mid F_{p}, \\ p \equiv \pm1 \pmod 5 & \Rightarrow p \mid F_{p-1}, \\ p \equiv \pm2 \pmod 5 &  \Rightarrow p \mid F_{p+1}.\end{cases}</math>

These cases can be combined into a single, non-[[piecewise]] formula, using the [[Legendre symbol]]:<ref>{{citation
 | last = Williams | first = H. C.
 | doi = 10.4153/CMB-1982-053-0 | doi-access=free
 | issue = 3
 | journal = [[Canadian Mathematical Bulletin]]
 | mr = 668957
 | pages = 366–70
 | title = A note on the Fibonacci quotient <math>F_{p-\varepsilon}/p</math>
 | volume = 25
 | year = 1982| hdl = 10338.dmlcz/137492
 | hdl-access = free
 }}. Williams calls this property "well known".</ref>
<math display=block>p \mid F_{p \,-~\! \left(\frac{5}{p}\right)}.</math>

=== Primality testing ===
The above formula can be used as a [[primality test]] in the sense that if
<math display=block>n \mid F_{n \,-~\! \left(\frac{5}{n}\right)},</math>
where the Legendre symbol has been replaced by the [[Jacobi symbol]], then this is evidence that {{mvar|n}} is a prime, and if it fails to hold, then {{mvar|n}} is definitely not a prime. If {{mvar|n}} is [[composite number|composite]] and satisfies the formula, then {{mvar|n}} is a ''Fibonacci pseudoprime''. When {{mvar|m}} is large{{snd}}say a 500-[[bit]] number{{snd}}then we can calculate {{math|''F''<sub>''m''</sub> (mod ''n'')}} efficiently using the matrix form. Thus

<math display=block> \begin{pmatrix} F_{m+1} & F_m \\ F_m & F_{m-1} \end{pmatrix} \equiv \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^m \pmod n.</math>
Here the matrix power {{math|''A''<sup>''m''</sup>}} is calculated using [[modular exponentiation]], which can be [[Modular exponentiation#Matrices|adapted to matrices]].<ref>''Prime Numbers'', Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.</ref>

=== Fibonacci primes ===
{{Main|Fibonacci prime}}

A ''Fibonacci prime'' is a Fibonacci number that is [[prime number|prime]]. The first few are:<ref>{{Cite OEIS|1=A005478|2=Prime Fibonacci numbers|mode=cs2}}</ref>

: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.<ref>{{citation
 | last = Diaconis
 | first = Persi
 | author-link = Persi Diaconis
 | editor1-last = Butler
 | editor1-first = Steve
 | editor1-link = Steve Butler (mathematician)
 | editor2-last = Cooper
 | editor2-first = Joshua
 | editor3-last = Hurlbert
 | editor3-first = Glenn
 | contribution = Probabilizing Fibonacci numbers
 | contribution-url = https://statweb.stanford.edu/~cgates/PERSI/papers/probabilizing-fibonacci.pdf
 | isbn = 978-1-107-15398-1
 | mr = 3821829
 | pages = 1–12
 | publisher = Cambridge University Press
 | title = Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham
 | year = 2018
 | access-date = 2022-11-23
 | archive-date = 2023-11-18
 | archive-url = https://web.archive.org/web/20231118192225/https://statweb.stanford.edu/~cgates/PERSI/papers/probabilizing-fibonacci.pdf
 | url-status = dead
 }}</ref>

{{math|''F''<sub>''kn''</sub>}} is divisible by {{math|''F''<sub>''n''</sub>}}, so, apart from {{math|1=''F''<sub>4</sub> = 3}}, any Fibonacci prime must have a prime index. As there are [[Arbitrarily large|arbitrarily long]] runs of [[composite number]]s, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

No Fibonacci number greater than {{math|1=''F''<sub>6</sub> = 8}} is one greater or one less than a prime number.<ref>{{Citation | first = Ross | last = Honsberger | title = Mathematical Gems III | journal = AMS Dolciani Mathematical Expositions | year = 1985 | isbn = 978-0-88385-318-4 | page = 133 | issue = 9}}</ref>

The only nontrivial [[square number|square]] Fibonacci number is 144.<ref>{{citation | last = Cohn | first = J. H. E. | doi = 10.1112/jlms/s1-39.1.537 | journal = The Journal of the London Mathematical Society | mr = 163867 | pages = 537–540 | title = On square Fibonacci numbers | volume = 39 | year = 1964}}</ref>  Attila Pethő proved in 2001 that there is only a finite number of [[perfect power]] Fibonacci numbers.<ref>{{Citation | first = Attila | last = Pethő | title = Diophantine properties of linear recursive sequences II | journal = Acta Mathematica Academiae Paedagogicae Nyíregyháziensis | volume = 17 | year = 2001 | pages = 81–96}}</ref> In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.<ref>{{Citation|first1=Y|last1=Bugeaud|first2=M|last2= Mignotte|first3=S|last3=Siksek|title = Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers | journal = Ann. Math.|volume = 2 | year = 2006 | pages = 969–1018 | issue = 163 | bibcode = 2004math......3046B | arxiv = math/0403046| doi = 10.4007/annals.2006.163.969|s2cid=10266596}}</ref>

1, 3, 21, and 55 are the only [[triangular number|triangular]] Fibonacci numbers, which was [[conjecture]]d by [[Verner Emil Hoggatt Jr.|Vern Hoggatt]] and proved by Luo Ming.<ref>{{Citation|first=Ming|last=Luo|title = On triangular Fibonacci numbers | journal = Fibonacci Quart. | volume = 27 | issue = 2 | year = 1989 | pages = 98–108 |doi=10.1080/00150517.1989.12429576 | url = https://www.fq.math.ca/Scanned/27-2/ming.pdf }}</ref>

No Fibonacci number can be a [[perfect number]].<ref name="Luca2000">{{citation | first=Florian | last=Luca | title=Perfect Fibonacci and Lucas numbers | journal=Rendiconti del Circolo Matematico di Palermo | year=2000 | volume=49 | issue=2 | pages=313–18 | doi=10.1007/BF02904236 | mr=1765401 | s2cid=121789033 | issn=1973-4409 }}</ref> More generally, no Fibonacci number other than 1 can be [[multiply perfect number|multiply perfect]],<ref name="BGLLHT2011">{{citation | first1=Kevin A. | last1=Broughan | first2=Marcos J. | last2=González | first3=Ryan H. | last3=Lewis | first4=Florian | last4=Luca | first5=V. Janitzio | last5=Mejía Huguet | first6=Alain | last6=Togbé | title=There are no multiply-perfect Fibonacci numbers | journal=Integers | year=2011 | volume=11a | page=A7 | url=https://math.colgate.edu/~integers/vol11a.html | mr=2988067 }}</ref> and no ratio of two Fibonacci numbers can be perfect.<ref name="LucaMH2010">{{citation | first1=Florian | last1=Luca | first2= V. Janitzio | last2=Mejía Huguet | title=On Perfect numbers which are ratios of two Fibonacci numbers | journal=Annales Mathematicae at Informaticae | year=2010 | volume=37 | pages=107–24 | url=http://ami.ektf.hu/index.php?vol=37 | mr=2753031 | issn=1787-6117 }}</ref>

=== Prime divisors ===
With the exceptions of 1, 8 and 144 ({{math|1=''F''<sub>1</sub> = ''F''<sub>2</sub>}}, {{math|''F''<sub>6</sub>}} and {{math|''F''<sub>12</sub>}}) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number ([[Carmichael's theorem]]).<ref>{{Citation | first = Ron | last = Knott | url = http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibtable.html | title = The Fibonacci numbers | publisher = Surrey | place = UK}}</ref> As a result, 8 and 144 ({{math|''F''<sub>6</sub>}} and {{math|''F''<sub>12</sub>}}) are the only Fibonacci numbers that are the product of other Fibonacci numbers.<ref>{{Cite OEIS|1=A235383|2=Fibonacci numbers that are the product of other Fibonacci numbers|mode=cs2}}</ref>

The divisibility of Fibonacci numbers by a prime {{mvar|p}} is related to the [[Legendre symbol]] <math>\bigl(\tfrac{p}{5}\bigr)</math> which is evaluated as follows:
<math display=block>\left(\frac{p}{5}\right) = \begin{cases} 0 & \text{if } p = 5\\ 1 & \text{if } p \equiv \pm 1 \pmod 5\\ -1 & \text{if } p \equiv \pm 2 \pmod 5.\end{cases}</math>

If {{mvar|p}} is a prime number then
<math display=block> F_p \equiv \left(\frac{p}{5}\right) \pmod p \quad \text{and}\quad F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p.</math><ref>{{Citation | first = Paulo | last = Ribenboim | author-link = Paulo Ribenboim | year = 1996 | title = The New Book of Prime Number Records | place = New York | publisher = Springer | isbn = 978-0-387-94457-9 | page = 64}}</ref>{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.25–28 | pp = 73–74}}

For example,
<math display=block>\begin{align}
\bigl(\tfrac{2}{5}\bigr) &= -1, &F_3  &= 2, &F_2&=1, \\
\bigl(\tfrac{3}{5}\bigr) &= -1, &F_4  &= 3,&F_3&=2, \\
\bigl(\tfrac{5}{5}\bigr) &= 0, &F_5  &= 5, \\
\bigl(\tfrac{7}{5}\bigr) &= -1, &F_8  &= 21,&F_7&=13, \\
\bigl(\tfrac{11}{5}\bigr)& = +1, &F_{10}&  = 55, &F_{11}&=89.
\end{align}</math>

It is not known whether there exists a prime {{mvar|p}} such that

<math display=block>F_{p\,-~\!\left(\frac{p}{5}\right)} \equiv 0 \pmod{p^2}.</math>

Such primes (if there are any) would be called [[Wall–Sun–Sun prime]]s.

Also, if {{math|''p'' ≠ 5}} is an odd prime number then:{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.28 | pp = 73–74}}
<math display=block>5 {F_{\frac{p \pm 1}{2}}}^2 \equiv \begin{cases}
\tfrac{1}{2} \left (5\bigl(\tfrac{p}{5}\bigr)\pm 5 \right ) \pmod p & \text{if } p \equiv 1 \pmod 4\\
\tfrac{1}{2} \left (5\bigl(\tfrac{p}{5}\bigr)\mp 3 \right ) \pmod p & \text{if } p \equiv 3 \pmod 4.
\end{cases}</math>

'''Example 1.''' {{math|1=''p'' = 7}}, in this case {{math|1=''p'' ≡ 3 (mod 4)}} and we have:
<math display=block>\bigl(\tfrac{7}{5}\bigr) = -1: \qquad \tfrac{1}{2}\left(5 \bigl(\tfrac{7}{5}\bigr)+3 \right ) =-1, \quad \tfrac{1}{2} \left(5\bigl(\tfrac{7}{5}\bigr)-3 \right )=-4.</math>
<math display=block>F_3=2 \text{ and } F_4=3.</math>
<math display=block>5{F_3}^2=20\equiv -1 \pmod {7}\;\;\text{ and }\;\;5{F_4}^2=45\equiv -4 \pmod {7}</math>

'''Example 2.''' {{math|1=''p'' = 11}}, in this case {{math|1=''p'' ≡ 3 (mod 4)}} and we have:
<math display=block>\bigl(\tfrac{11}{5}\bigr) = +1: \qquad  \tfrac{1}{2}\left( 5\bigl(\tfrac{11}{5}\bigr)+3 \right)=4, \quad \tfrac{1}{2} \left(5\bigl(\tfrac{11}{5}\bigr)- 3 \right)=1.</math>
<math display=block>F_5=5 \text{ and } F_6=8.</math>
<math display=block>5{F_5}^2=125\equiv 4 \pmod {11} \;\;\text{ and }\;\;5{F_6}^2=320\equiv 1 \pmod {11}</math>

'''Example 3.''' {{math|1=''p'' = 13}}, in this case {{math|1=''p'' ≡ 1 (mod 4)}} and we have:
<math display=block>\bigl(\tfrac{13}{5}\bigr) = -1: \qquad \tfrac{1}{2}\left(5\bigl(\tfrac{13}{5}\bigr)-5 \right) =-5, \quad \tfrac{1}{2}\left(5\bigl(\tfrac{13}{5}\bigr)+ 5 \right)=0.</math>
<math display=block>F_6=8 \text{ and } F_7=13.</math>
<math display=block>5{F_6}^2=320\equiv -5 \pmod {13} \;\;\text{ and }\;\;5{F_7}^2=845\equiv 0 \pmod {13}</math>

'''Example 4.''' {{math|1=''p'' = 29}}, in this case {{math|1=''p'' ≡ 1 (mod 4)}} and we have:
<math display=block>\bigl(\tfrac{29}{5}\bigr) = +1: \qquad \tfrac{1}{2}\left(5\bigl(\tfrac{29}{5}\bigr)-5 \right)=0, \quad \tfrac{1}{2}\left(5\bigl(\tfrac{29}{5}\bigr)+5 \right)=5.</math>
<math display=block>F_{14}=377 \text{ and } F_{15}=610.</math>
<math display=block>5{F_{14}}^2=710645\equiv 0 \pmod {29} \;\;\text{ and }\;\;5{F_{15}}^2=1860500\equiv 5 \pmod {29}</math>

For odd {{mvar|n}}, all odd prime divisors of {{math|''F''<sub>''n''</sub>}} are congruent to 1 modulo 4, implying that all odd divisors of {{math|1=''F''<sub>''n''</sub>}} (as the products of odd prime divisors) are congruent to 1 modulo 4.{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.27 | p = 73}}

For example,
<math display=block>F_1 = 1,\ F_3 = 2,\ F_5 = 5,\ F_7 = 13,\ F_9 = {\color{Red}34} = 2 \cdot 17,\ F_{11} = 89,\ F_{13} = 233,\ F_{15} = {\color{Red}610} = 2 \cdot 5 \cdot 61.</math>

All known factors of Fibonacci numbers {{math|''F''(''i'')}} for all {{math|''i'' < 50000}} are collected at the relevant repositories.<ref>{{Citation | url = https://mersennus.net/fibonacci/ | title = Fibonacci and Lucas factorizations | publisher = Mersennus}} collects all known factors of {{math|''F''(''i'')}} with {{math|''i'' < 10000}}</ref><ref>{{Citation | url =http://fibonacci.redgolpe.com/ | title = Factors of Fibonacci and Lucas numbers | publisher = Red golpe}} collects all known factors of {{math|''F''(''i'')}} with {{math|10000 < ''i'' < 50000}}</ref>

=== Periodicity modulo ''n'' ===
{{Main|Pisano period}}

If the members of the Fibonacci sequence are taken mod&nbsp;{{mvar|n}}, the resulting sequence is [[periodic sequence|periodic]] with period at most&nbsp;{{math|6''n''}}.<ref>{{Citation | title = Problems and Solutions: Solutions: E3410 | last1 = Freyd | first1 = Peter | last2 = Brown | first2 = Kevin S. | journal = The American Mathematical Monthly | volume = 99 | issue = 3 | pages = 278–79 |date= 1993 | doi=10.2307/2325076| jstor = 2325076 }}</ref> The lengths of the periods for various {{mvar|n}} form the so-called [[Pisano period]]s.<ref>{{Cite OEIS|1=A001175|2=Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n|mode=cs2}}</ref> Determining a general formula for the Pisano periods is an [[open problem]], which includes as a subproblem a special instance of the problem of finding the [[multiplicative order]] of a [[modular arithmetic|modular integer]] or of an element in a [[finite field]]. However, for any particular {{mvar|n}}, the Pisano period may be found as an instance of [[cycle detection]].

== Generalizations ==
{{Main|Generalizations of Fibonacci numbers}}

The Fibonacci sequence is one of the simplest and earliest known sequences defined by a [[recurrence relation]], and specifically by a linear [[difference equation]]. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a [[linear recurrence with constant coefficients|homogeneous linear difference equation with constant coefficients]].

Some specific examples that are close, in some sense, to the Fibonacci sequence include:
* Generalizing the index to negative integers to produce the [[negafibonacci]] numbers.
* Generalizing the index to [[real number]]s using a modification of Binet's formula.<ref name="MathWorld" />
* Starting with other integers. [[Lucas number]]s have {{math|1=''L''<sub>1</sub> = 1}}, {{math|1=''L''<sub>2</sub> = 3}}, and {{math|1=''L<sub>n</sub>'' = ''L''<sub>''n''−1</sub> + ''L''<sub>''n''−2</sub>}}. [[Primefree sequence]]s use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.
* Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The [[Pell number]]s have {{math|1=''P<sub>n</sub>'' = 2''P''<sub>''n''−1</sub> + ''P''<sub>''n''−2</sub>}}. If the coefficient of the preceding value is assigned a variable value {{mvar|x}}, the result is the sequence of [[Fibonacci polynomials]].
* Not adding the immediately preceding numbers. The [[Padovan sequence]] and [[Perrin number]]s have {{math|1=''P''(''n'') = ''P''(''n'' − 2) + ''P''(''n'' − 3)}}.
* Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as ''n-Step Fibonacci numbers''.<ref>{{citation
 | last1 = Lü | first1 = Kebo
 | last2 = Wang | first2 = Jun
 | journal = Utilitas Mathematica
 | mr = 2278830
 | pages = 169–177
 | title = {{mvar|k}}-step Fibonacci sequence modulo {{mvar|m}}
 | url = https://utilitasmathematica.com/index.php/Index/article/view/410
 | volume = 71
 | year = 2006}}</ref>

== Applications ==

=== Mathematics ===

[[File:Pascal triangle fibonacci.svg|thumb|upright=1.2|The Fibonacci numbers are the sums of the diagonals (shown in red) of a left-justified [[Pascal's triangle]].]]
The Fibonacci numbers occur as the sums of [[binomial coefficient]]s in the "shallow" diagonals of [[Pascal's triangle]]:{{Sfn | Lucas | 1891 | p = 7}}
<math display=block>F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}.</math>
This can be proved by expanding the generating function
<math display=block>\frac{x}{1-x-x^2} = x + x^2(1+x) + x^3(1+x)^2 + \dots + x^{k+1}(1+x)^k + \dots = \sum\limits_{n=0}^\infty F_n x^n</math>
and collecting like terms of <math>x^n</math>.

To see how the formula is used, we can arrange the sums by the number of terms present:
:{|
| {{math|5}}
| {{math|1== 1+1+1+1+1}}
|-
|
| {{math|1== 2+1+1+1}}
| {{math|1== 1+2+1+1}}
| {{math|1== 1+1+2+1}}
| {{math|1== 1+1+1+2}}
|-
|
| {{math|1== 2+2+1}}
| {{math|1== 2+1+2}}
| {{math|1== 1+2+2}}
|}

which is <math>\textstyle \binom{5}{0}+\binom{4}{1}+\binom{3}{2}</math>, where we are choosing the positions of {{mvar|k}} twos from {{math|''n''−''k''−1}} terms.

[[File:Fibonacci climbing stairs.svg|thumb|right|Use of the Fibonacci sequence to count {{nowrap|{1,&thinsp;2}-restricted}} compositions]]
These numbers also give the solution to certain enumerative problems,<ref>{{citation|last=Stanley|first=Richard|title=Enumerative Combinatorics I (2nd ed.)|year=2011|publisher=Cambridge Univ. Press|isbn=978-1-107-60262-5|page=121, Ex 1.35}}</ref> the most common of which is that of counting the number of ways of writing a given number {{mvar|n}} as an ordered sum of 1s and 2s (called [[composition (combinatorics)#Number of compositions|compositions]]); there are {{math|''F''<sub>''n''+1</sub>}} ways to do this (equivalently, it's also the number of [[domino tiling]]s of the <math>2\times n</math> rectangle). For example, there are {{math|1=''F''<sub>5+1</sub> = ''F''<sub>6</sub> = 8}} ways one can climb a staircase of 5 steps, taking one or two steps at a time:
:{|
| {{math|5}}
| {{math|1== 1+1+1+1+1}}
| {{math|1== 2+1+1+1}}
| {{math|1== 1+2+1+1}}
| {{math|1== 1+1+2+1}}
| {{math|1== 2+2+1}}
|-
|
| {{math|1== 1+1+1+2}}
| {{math|1== 2+1+2}}
| {{math|1== 1+2+2}}
|}
The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied [[recursion|recursively]] until a single step, of which there is only one way to climb.

The Fibonacci numbers can be found in different ways among the set of [[binary numeral system|binary]] [[String (computer science)|strings]], or equivalently, among the [[subset]]s of a given set.
* The number of binary strings of length {{mvar|n}} without consecutive {{mono|1}}s is the Fibonacci number {{math|''F''<sub>''n''+2</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''<sub>6</sub> = 8}} without consecutive {{math|1}}s—they are {{mono|0000}}, {{mono|0001}}, {{mono|0010}}, {{mono|0100}}, {{mono|0101}}, {{mono|1000}}, {{mono|1001}}, and {{mono|1010}}. Such strings are the binary representations of [[Fibbinary number]]s. Equivalently, {{math|''F''<sub>''n''+2</sub>}} is the number of subsets {{mvar|S}} of {{math|{{mset|1, ..., ''n''}}}} without consecutive integers, that is, those {{mvar|S}} for which {{math|{{mset|''i'', ''i'' + 1}} ⊈ ''S''}} for every {{mvar|i}}. A [[bijection]] with the sums to {{math|''n''+1}} is to replace 1 with {{mono|0}} and 2 with {{mono|10}}, and drop the last zero.
* The number of binary strings of length {{mvar|n}} without an odd number of consecutive {{mono|1}}s is the Fibonacci number {{math|''F''<sub>''n''+1</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''<sub>5</sub> = 5}} without an odd number of consecutive {{mono|1}}s—they are {{mono|0000}}, {{mono|0011}}, {{mono|0110}}, {{mono|1100}}, {{mono|1111}}. Equivalently, the number of subsets {{mvar|S}} of {{math|{{mset|1, ..., ''n''}}}} without an odd number of consecutive integers is {{math|''F''<sub>''n''+1</sub>}}. A bijection with the sums to {{mvar|n}} is to replace 1 with {{mono|0}} and 2 with {{mono|11}}.
* The number of binary strings of length {{mvar|n}} without an even number of consecutive {{mono|0}}s or {{mono|1}}s is {{math|2''F''<sub>''n''</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=2''F''<sub>4</sub> = 6}} without an even number of consecutive {{mono|0}}s or {{mono|1}}s—they are {{mono|0001}}, {{mono|0111}}, {{mono|0101}}, {{mono|1000}}, {{mono|1010}}, {{mono|1110}}. There is an equivalent statement about subsets.
* [[Yuri Matiyasevich]] was able to show that the Fibonacci numbers can be defined by a [[Diophantine equation]], which led to [[Matiyasevich's theorem|his solving]] [[Hilbert's tenth problem]].<ref>{{citation|title=Review of Yuri V. Matiyasevich, ''Hibert's Tenth Problem''|journal=Modern Logic|first=Valentina|last=Harizanov|author-link=Valentina Harizanov|volume=5|issue=3|year=1995|pages=345–55|url=https://projecteuclid.org/euclid.rml/1204900767}}</ref>
* The Fibonacci numbers are also an example of a [[complete sequence]]. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
* Moreover, every positive integer can be written in a unique way as the sum of ''one or more'' distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as [[Zeckendorf's theorem]], and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its [[Fibonacci coding]].
* Starting with 5, every second Fibonacci number is the length of the [[hypotenuse]] of a [[right triangle]] with integer sides, or in other words, the largest number in a [[Pythagorean triple]], obtained from the formula <math display=block>(F_n F_{n+3})^2 + (2 F_{n+1}F_{n+2})^2 = {F_{2 n+3}}^2.</math> The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ...&thinsp;. The middle side of each of these triangles is the sum of the three sides of the preceding triangle.<ref>{{citation
 | last = Pagni | first = David
 | date = September 2001
 | issue = 4
 | journal = Mathematics in School
 | jstor = 30215477
 | pages = 39–40
 | title = Fibonacci Meets Pythagoras
 | volume = 30}}</ref>
* The [[Fibonacci cube]] is an [[undirected graph]] with a Fibonacci number of nodes that has been proposed as a [[network topology]] for [[parallel computing]].
* Fibonacci numbers appear in the [[ring lemma]], used to prove connections between the [[circle packing theorem]] and [[conformal map]]s.<ref>{{citation|last=Stephenson|first=Kenneth|isbn=978-0-521-82356-2|mr=2131318|publisher=Cambridge University Press|title=Introduction to Circle Packing: The Theory of Discrete Analytic Functions|title-link=Introduction to Circle Packing|year=2005}}; see especially Lemma 8.2 (Ring Lemma), [https://books.google.com/books?id=38PxEmKKhysC&pg=PA73 pp. 73–74], and Appendix B, The Ring Lemma, pp. 318–321.</ref>

=== Computer science ===
[[File:Fibonacci Tree 6.svg|thumb|upright=1.2|Fibonacci tree of height 6. [[AVL tree#Balance factor|Balance factor]]s green; heights red.<br />The keys in the left spine are Fibonacci numbers.]]

* The Fibonacci numbers are important in [[Analysis of algorithms|computational run-time analysis]] of [[Euclidean algorithm|Euclid's algorithm]] to determine the [[greatest common divisor]] of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.<ref>{{Citation| first= Donald E |last= Knuth| author-link= Donald Knuth | year =1997|title=The Art of Computer Programming | volume = 1: Fundamental Algorithms|edition= 3rd | publisher = Addison–Wesley |isbn=978-0-201-89683-1 | page = 343}}</ref>
* Fibonacci numbers are used in a polyphase version of the [[merge sort]] algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers—by dividing the list so that the two parts have lengths in the approximate proportion {{mvar|φ}}. A tape-drive implementation of the [[polyphase merge sort]] was described in ''[[The Art of Computer Programming]]''.
* {{anchor|Fibonacci Tree}}A Fibonacci tree is a [[binary tree]] whose child trees (recursively) differ in [[Tree height|height]] by exactly 1. So it is an [[AVL tree]], and one with the fewest nodes for a given height—the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.<ref>{{citation|last1=Adelson-Velsky|first1=Georgy|last2=Landis|first2=Evgenii|year=1962|title=An algorithm for the organization of information|journal=[[Proceedings of the USSR Academy of Sciences]]|volume=146|pages=263–266|language=ru}} [https://zhjwpku.com/assets/pdf/AED2-10-avl-paper.pdf English translation] by Myron J. Ricci in ''Soviet Mathematics - Doklady'', 3:1259–1263, 1962.</ref>
* Fibonacci numbers are used by some [[pseudorandom number generator]]s.<!-- Knuth vol. 2 -->
* Fibonacci numbers arise in the analysis of the [[Fibonacci heap]] data structure.
* A one-dimensional optimization method, called the [[Fibonacci search technique]], uses Fibonacci numbers.<ref>{{Citation| first1 = M | last1 = Avriel | first2 = DJ | last2 = Wilde | title= Optimality of the Symmetric Fibonacci Search Technique |journal=Fibonacci Quarterly|year=1966 |issue=3 |pages= 265–69| doi = 10.1080/00150517.1966.12431364 }}</ref>
* The Fibonacci number series is used for optional [[lossy compression]] in the [[Interchange File Format|IFF]] [[8SVX]] audio file format used on [[Amiga]] computers. The number series [[companding|compands]] the original audio wave similar to logarithmic methods such as [[μ-law]].<ref>{{Citation | title = Amiga ROM Kernel Reference Manual | publisher = Addison–Wesley | year = 1991}}</ref><ref>{{Citation | url = https://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression | contribution = IFF | title = Multimedia Wiki}}</ref>
* Some Agile teams use a modified series called the "Modified Fibonacci Series" in [[planning poker]], as an estimation tool. Planning Poker is a formal part of the [[Scaled agile framework|Scaled Agile Framework]].<ref>{{citation|author=Dean Leffingwell |url=https://www.scaledagileframework.com/story/ |title=Story |publisher=Scaled Agile Framework |date=2021-07-01 |access-date=2022-08-15}}</ref>
* [[Fibonacci coding]]
* [[Negafibonacci coding]]

=== Nature ===
{{Further|Patterns in nature}}
{{see also|Golden ratio#Nature}}
[[File:FibonacciChamomile.PNG|thumb|[[Yellow chamomile]] head showing the arrangement in 21 (blue) and 13 (cyan) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.]]

Fibonacci sequences appear in biological settings,<ref>{{Citation |first1=S |last1=Douady |first2=Y |last2=Couder |title=Phyllotaxis as a Dynamical Self Organizing Process |journal=Journal of Theoretical Biology |year=1996 |issue=3 |pages=255–74 |url=http://www.math.ntnu.no/~jarlet/Douady96.pdf |doi=10.1006/jtbi.1996.0026 |volume=178 |url-status=dead |archive-url=https://web.archive.org/web/20060526054108/http://www.math.ntnu.no/~jarlet/Douady96.pdf |archive-date=2006-05-26 }}</ref> such as branching in trees, [[Phyllotaxis|arrangement of leaves on a stem]], the fruitlets of a [[pineapple]],<ref>{{Citation | first1=Judy |last1=Jones | first2=William | last2=Wilson |title=An Incomplete Education |publisher=Ballantine Books |year=2006 |isbn=978-0-7394-7582-9 |page=544 |chapter=Science}}</ref> the flowering of [[artichoke]], the leaves of the spiral aloe<ref>{{Cite web |title=The Wonder of Fibonacci in our Gardens {{!}} UC Master Gardeners of San Mateo & San Francisco Counties |url=https://ucanr.edu/blog/uc-master-gardeners-san-mateo-san-francisco-counties/article/wonder-fibonacci-our-gardens |access-date=2025-11-18 |website=ucanr.edu |language=en}}</ref> (Aloe polyphylla), the arrangement of a [[pine cone]],<ref>{{Citation| first=A | last=Brousseau |title=Fibonacci Statistics in Conifers | journal=[[Fibonacci Quarterly]] |year=1969 |issue=7 |pages=525–32}}</ref> and the family tree of [[honeybee]]s.<ref>{{citation|url = https://www.cs4fn.org/maths/bee-davinci.php |work = Maths | publisher = Computer Science For Fun: CS4FN |title = Marks for the da Vinci Code: B–}}</ref><ref>{{Citation|first1=T.C.|last1=Scott|first2=P.|last2=Marketos| url = http://www-history.mcs.st-andrews.ac.uk/Publications/fibonacci.pdf | title = On the Origin of the Fibonacci Sequence | publisher = [[MacTutor History of Mathematics archive]], University of St Andrews| date = March 2014}}</ref> [[Kepler]] pointed out the presence of the Fibonacci sequence in nature, using it to explain the ([[golden ratio]]-related) [[pentagon]]al form of some flowers.{{sfn|Livio|2003|p=110}} Field [[Leucanthemum vulgare|daisies]] most often have petals in counts of Fibonacci numbers.{{sfn|Livio|2003|pp=112–13}} In 1830, [[Karl Friedrich Schimper]] and [[Alexander Braun]] discovered that the [[Parastichy|parastichies]] (spiral [[phyllotaxis]]) of plants were frequently expressed as fractions involving Fibonacci numbers.<ref>{{Citation |first =Franck |last = Varenne |title = Formaliser le vivant - Lois, Théories, Modèles | access-date = 2022-10-30| url = https://www.numilog.com/LIVRES/ISBN/9782705670894.Livre | page = 28 | date = 2010| isbn = 9782705678128|publisher = Hermann|quote = En 1830, K. F. Schimper et A. Braun [...]. Ils montraient que si l'on représente cet angle de divergence par une fraction reflétant le nombre de tours par feuille ([...]), on tombe régulièrement sur un des nombres de la suite de Fibonacci pour le numérateur [...].|language = fr}}</ref>

[[Przemysław Prusinkiewicz]] advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on [[free group]]s, specifically as certain [[L-system|Lindenmayer grammars]].<ref>{{Citation|first1 = Przemyslaw |last1 = Prusinkiewicz | first2 = James | last2 = Hanan| title = Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics) |publisher= [[Springer Science+Business Media|Springer-Verlag]] |year=1989 |isbn=978-0-387-97092-9}}</ref>

[[File:SunflowerModel.svg|thumb|Illustration of Vogel's model for {{math|''n'' {{=}} 1 ... 500}}]]
A model for the pattern of [[floret]]s in the head of a [[sunflower]] was proposed by {{ill|Helmut Vogel|de|Helmut Vogel (Physiker)}} in 1979.<ref>{{Citation | last =Vogel | first =Helmut | title =A better way to construct the sunflower head | journal = Mathematical Biosciences | issue =3–4 | pages = 179–89 | year = 1979 | doi = 10.1016/0025-5564(79)90080-4 | volume = 44}}</ref>  This has the form

<math display=block>\theta = \frac{2\pi}{\varphi^2} n,\  r = c \sqrt{n}</math>

where {{mvar|n}} is the index number of the floret and {{mvar|c}} is a constant scaling factor; the florets thus lie on [[Fermat's spiral]]. The divergence [[angle]], approximately 137.51°, is the [[golden angle]], dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form {{math|''F''(&thinsp;''j''):''F''(&thinsp;''j'' + 1)}}, the nearest neighbors of floret number {{mvar|n}} are those at {{math|''n'' ± ''F''(&thinsp;''j'')}} for some index {{mvar|j}}, which depends on {{mvar|r}}, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,{{sfn|Livio|2003|p=112}} typically counted by the outermost range of radii.<ref>{{Citation | last1 = Prusinkiewicz | first1 = Przemyslaw | author1-link = Przemyslaw Prusinkiewicz | author2-link = Aristid Lindenmayer | last2 = Lindenmayer | first2 = Aristid | title = The Algorithmic Beauty of Plants | publisher = Springer-Verlag | year = 1990 | pages = [https://archive.org/details/algorithmicbeaut0000prus/page/101 101–107] | chapter = 4 | chapter-url = https://algorithmicbotany.org/papers/#webdocs | isbn = 978-0-387-97297-8 | url = https://archive.org/details/algorithmicbeaut0000prus/page/101 }}</ref>

Fibonacci numbers also appear in the ancestral pedigrees of [[bee]]s (which are [[haplodiploid]]s), according to the following rules:
* If an egg is laid but not fertilized, it produces a male (or [[Drone (bee)|drone bee]] in honeybees).
* If, however, an egg is fertilized, it produces a female.

Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, {{math|''F''<sub>''n''</sub>}}, is the number of female ancestors, which is {{math|''F''<sub>''n''−1</sub>}}, plus the number of male ancestors, which is {{math|''F''<sub>''n''−2</sub>}}.<ref>{{Citation | url = https://www.fq.math.ca/Scanned/1-1/basin.pdf | title = The Fibonacci sequence as it appears in nature | journal = The Fibonacci Quarterly | volume = 1 | number = 1 | pages = 53–56 | year = 1963| doi = 10.1080/00150517.1963.12431602 | last1 = Basin | first1 = S. L. }}</ref><ref>Yanega, D. 1996. Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae). J. Kans. Ent. Soc. 69 Suppl.: 98-115.</ref> This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

[[File:X chromosome ancestral line Fibonacci sequence.svg|thumb|upright=1.2|The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships".<ref name="xcs"/>)]]

It has similarly been noticed that the number of possible ancestors on the human [[X chromosome]] inheritance line at a given ancestral generation also follows the Fibonacci sequence.<ref name="xcs">{{citation|last=Hutchison|first=Luke|date=September 2004|title=Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships|url=http://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|journal=Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04)|access-date=2016-09-03|archive-date=2020-09-25|archive-url=https://web.archive.org/web/20200925132536/https://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|url-status=dead}}</ref> A male individual has an X chromosome, which he received from his mother, and a [[Y chromosome]], which he received from his father. The male counts as the "origin" of his own X chromosome (<math>F_1=1</math>), and at his parents' generation, his X chromosome came from a single parent {{nowrap|(<math>F_2=1</math>)}}. The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_3=2</math>)}}. The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_4=3</math>)}}. Five great-great-grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_5=5</math>)}}, etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a [[Founder effect|population founder]] appears on all lines of the genealogy.)

===Other===
* In [[optics]], when a beam of light shines at an angle through two stacked transparent plates of different materials of different [[refractive index]]es, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have {{mvar|k}} reflections, for {{math|''k'' > 1}}, is the {{mvar|k}}-th Fibonacci number. (However, when {{math|1=''k'' = 1}}, there are three reflection paths, not two, one for each of the three surfaces.){{sfn|Livio|2003|pp=98–99}}
* [[Fibonacci retracement]] levels are widely used in [[technical analysis]] for financial market trading.
* Since the [[conversion of units|conversion]] factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a [[radix]] 2 number [[processor register|register]] in [[golden ratio base]] {{mvar|φ}} being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.<ref>{{Citation | url = https://www.encyclopediaofmath.org/index.php/Zeckendorf_representation | contribution = Zeckendorf representation | title = Encyclopedia of Math}}</ref>
* The measured values of voltages and currents in the infinite resistor chain circuit (also called the [[resistor ladder]] or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.<ref>{{citation
 | last1 = Patranabis | first1 = D.
 | last2 = Dana | first2 = S. K.
 | date = December 1985
 | doi = 10.1109/tim.1985.4315428
 | issue = 4
 | journal = [[IEEE Transactions on Instrumentation and Measurement]]
 | pages = 650–653
 | title = Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers
 | volume = IM-34| bibcode = 1985ITIM...34..650P
 | s2cid = 35413237
 }}</ref>
* Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of [[economics]].<ref name="Brasch et al. 2012">{{Citation| first1 =T. von | last1 = Brasch | first2 = J. | last2 = Byström | first3 = L.P. | last3 = Lystad| title= Optimal Control and the Fibonacci Sequence |journal = Journal of Optimization Theory and Applications |year=2012 |issue=3 |pages= 857–78 |doi = 10.1007/s10957-012-0061-2
|volume=154 | hdl = 11250/180781 | s2cid = 8550726 | url = https://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-24073 | hdl-access = free }}</ref> In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.
* [[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 ==
* {{annotated link|The Fibonacci Association}}
* {{annotated link|Fibonacci numbers in popular culture}}
* {{annotated link|Fibonacci word}}
* {{annotated link|Random Fibonacci sequence}}
* {{annotated link|Wythoff array}}
* {{annotated link|International Conference on Fibonacci Numbers and their Applications}}

== References ==
=== Explanatory footnotes ===
{{Notelist}}

=== Citations ===
{{Reflist}}

===Works cited===
* {{Citation | title= Strange Curves, Counting Rabbits, and Other Mathematical Explorations | first= Keith M | last = Ball |publisher= [[Princeton University Press]]| place= Princeton, NJ | year= 2003 | chapter= 8: Fibonacci's Rabbits Revisited |isbn= 978-0-691-11321-0}}.
* {{Citation |title= The Art of Proof: Basic Training for Deeper Mathematics |first1= Matthias |last1= Beck |first2 = Ross |last2=Geoghegan |publisher=Springer |place=New York |year= 2010 |isbn=978-1-4419-7022-0}}.
* {{Citation |title=A Walk Through Combinatorics |edition= 3rd |first= Miklós |last= Bóna |author-link=Miklós Bóna |publisher= World Scientific | place=New Jersey |year= 2011 |isbn= 978-981-4335-23-2}}.
* {{anchor|Borwein}}{{Citation
  | last1 =Borwein
  | first1 =Jonathan M.
  | author-link =Jonathan Borwein
  | author-link2=Peter Borwein|first2=Peter B.|last2= Borwein
  | title =Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
  | pages =91–101
  | publisher =Wiley
  | date=July 1998
  | url =http://www.wiley.com/WileyCDA/WileyTitle/productCd-047131515X.html
  | isbn = 978-0-471-31515-5 }}
* {{Citation | first = Franz | last = Lemmermeyer | year = 2000 | title = Reciprocity Laws: From Euler to Eisenstein | series = Springer Monographs in Mathematics | place = New York | publisher = Springer | isbn = 978-3-540-66957-9}}.
* {{citation | last = Livio | first = Mario | author-link = Mario Livio | title = The Golden Ratio: The Story of Phi, the World's Most Astonishing Number | url = https://books.google.com/books?id=bUARfgWRH14C | orig-year = 2002 | edition = First trade paperback | year = 2003 | publisher = [[Random House|Broadway Books]] | location = New York City | isbn = 0-7679-0816-3 }}
* {{Citation |title=Théorie des nombres |first= Édouard |last= Lucas |publisher= Gauthier-Villars|year= 1891 | volume = 1 | language = fr | place = Paris | url = https://archive.org/details/thoriedesnombr01lucauoft}}.
* {{Citation | first = L. E. | last = Sigler | title = Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation |series=Sources and Studies in the History of Mathematics and Physical Sciences | publisher=Springer | year=2002 | isbn=978-0-387-95419-6}}

== External links ==
{{Wikiquote}}
{{Wikibooks|Fibonacci number program}}
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* {{YouTube|id=hbUQlrLDAgw|title=Fibonacci Sequence and Golden Ratio: Mathematics in the Modern World - Mathuklasan with Sir Ram}} - animation of sequence, spiral, golden ratio, rabbit pair growth. Examples in art, music, architecture, nature, and astronomy
* [https://www.mathpages.com/home/kmath078/kmath078.htm Periods of Fibonacci Sequences Mod m] at MathPages
* [http://www.physorg.com/news97227410.html Scientists find clues to the formation of Fibonacci spirals in nature]
* {{In Our Time|Fibonacci Sequence|b008ct2j|Fibonacci_Sequence}}
* {{springer|title=Fibonacci numbers|id=p/f040020}}

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