Fibonacci sequence#Binet's formula

File:Fibonacci Spiral.svg created by drawing circular arcs 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}}

$$F\_0=0,\backslash quad\; F\_1=\; 1,$$

and

$$F\_n=F\_\{n-1\}\; +\; F\_\{n-2\}$$

for {{math|n > 1}}.

Under some older definitions, the value $F\_0\; =\; 0$ is omitted, so that the sequence starts with $F\_1=F\_2=1,$ and the recurrence $F\_n=F\_\{n-1\}\; +\; F\_\{n-2\}$ is valid for {{math|n > 2}}.{{Sfn | Beck | Geoghegan | 2010}}{{Sfn | Bóna | 2011 | p=180}}

The first 20 Fibonacci numbers {{math|F_n}} are:

:

{{see also|Golden ratio#History}}

File:Fibonacci Sanskrit prosody.svg

The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.{{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}}{{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 ... both Gopala (before 1135 AD) and Hemachandra (c. 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) ...}}{{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_m+1}}.{{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)}}

Knowledge of the Fibonacci sequence was expressed as early as Pingala ({{circa}} 450 BC–200 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_m+1}}) is obtained by adding one [S] to the {{math|F_m}} cases and one [L] to the {{math|F_m−1}} cases.{{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}} Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).{{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}}

However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):{{sfn|Livio|2003|p=197}}

Variations of two earlier meters [is the variation] ... 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] ... 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 morae [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" {{Citation|last=Velankar|first=HD|year=1962|title='Vṛttajātisamuccaya' of kavi Virahanka|publisher=Rajasthan Oriental Research Institute|location=Jodhpur|page=101}}}}

Hemachandra (c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."{{sfn|Livio|2003|p=197–198}}{{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=4 January 2019}}

File:Liber abbaci magliab f124r.jpg's {{lang|la|Liber Abaci}} from the Biblioteca Nazionale di Firenze showing (in box on right) 13 entries of the Fibonacci sequence:
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}}{{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=28 November 2018}} where it is used to calculate the growth of rabbit populations.{{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}} Fibonacci considers the growth of an idealized (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 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.{{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}}

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.{{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}}

File:Fibonacci Rabbits.svg: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At the end of the nth month, the number of pairs is equal to F_n.]]

{{clear|left}}

{{main|Golden ratio}}

Like every sequence defined by a homogeneous linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression.{{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] 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:{{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 }}

F_n = \frac{\varphi^n-\psi^n}{\varphi-\psi} = \frac{\varphi^n-\psi^n}{\sqrt 5},

where

\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\ldots

is the golden ratio, and $\backslash psi$ is its conjugate:{{Sfn | Ball | 2003 | p = 156}}

\psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = - {1 \over \varphi} \approx -0.61803\,39887\ldots.

Since $\backslash psi\; =\; -\backslash varphi^\{-1\}$, this formula can also be written as

F_n = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt 5} = \frac{\varphi^n - (-\varphi)^{-n}}{2\varphi - 1}.

To see the relation between the sequence and these constants,{{Sfn | Ball | 2003 | pp = 155–156}} note that $\backslash varphi$ and $\backslash psi$ are both solutions of the equation $x^2\; =\; x\; +\; 1$ and thus $x^n\; =\; x^\{n-1\}\; +\; x^\{n-2\},$ so the powers of $\backslash varphi$ and $\backslash psi$ satisfy the Fibonacci recursion. In other words,

$$\backslash begin\{align\}$$

\varphi^n &= \varphi^{n-1} + \varphi^{n-2}, \\[3mu]

\psi^n &= \psi^{n-1} + \psi^{n-2}.

\end{align}

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

$$U\_n=a\; \backslash varphi^n\; +\; b\; \backslash psi^n$$

satisfies the same recurrence,

$$\backslash begin\{align\}$$

U_n &= a\varphi^n + b\psi^n \\[3mu]

&= a(\varphi^{n-1} + \varphi^{n-2}) + b(\psi^{n-1} + \psi^{n-2}) \\[3mu]

&= a\varphi^{n-1} + b\psi^{n-1} + a\varphi^{n-2} + b\psi^{n-2} \\[3mu]

&= U_{n-1} + U_{n-2}.

\end{align}

If {{mvar|a}} and {{mvar|b}} are chosen so that {{math|1=U₀ = 0}} and {{math|1=U₁ = 1}} then the resulting sequence {{math|U_n}} must be the Fibonacci sequence. This is the same as requiring {{mvar|a}} and {{mvar|b}} satisfy the system of equations:

\left\{\begin{align} a + b &= 0 \\ \varphi a + \psi b &= 1\end{align}\right.

which has solution

a = \frac{1}{\varphi-\psi} = \frac{1}{\sqrt 5},\quad b = -a,

producing the required formula.

Taking the starting values {{math|U₀}} and {{math|U₁}} to be arbitrary constants, a more general solution is:

$$U\_n\; =\; a\backslash varphi^n\; +\; b\backslash psi^n$$

where

$$\backslash begin\{align\}$$

a&=\frac{U_1-U_0\psi}{\sqrt 5}, \\[3mu]

b&=\frac{U_0\varphi-U_1}{\sqrt 5}.

\end{align}

Since

for all {{math|n ≥ 0}}, the number {{math|F_n}} is the closest integer to $\backslash frac\{\backslash varphi^n\}\{\backslash sqrt\; 5\}$. Therefore, it can be found by rounding, using the nearest integer function:

$$F\_n=\backslash left\backslash lfloor\backslash frac\{\backslash varphi^n\}\{\backslash sqrt\; 5\}\backslash right\backslash rceil,\backslash \; n\; \backslash geq\; 0.$$

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}}:

$$n(F)\; =\; \backslash left\backslash lfloor\; \backslash log\_\backslash varphi\; \backslash sqrt\{5\}F\backslash right\backslash rceil,\backslash \; F\; \backslash geq\; 1.$$

Instead using the floor function gives the largest index of a Fibonacci number that is not greater than {{mvar|F}}:

$$n\_\{\backslash mathrm\{largest\}\}(F)\; =\; \backslash left\backslash lfloor\; \backslash log\_\backslash varphi\; \backslash sqrt\{5\}(F+1/2)\backslash right\backslash rfloor,\backslash \; F\; \backslash geq\; 0,$$

where $\backslash log\_\backslash varphi(x)\; =\; \backslash ln(x)/\backslash ln(\backslash varphi)\; =\; \backslash log\_\{10\}(x)/\backslash log\_\{10\}(\backslash varphi)$, $\backslash ln(\backslash varphi)\; =\; 0.481211\backslash ldots$,{{Cite OEIS|1=A002390|2=Decimal expansion of natural logarithm of golden ratio|mode=cs2}} and $\backslash log\_\{10\}(\backslash varphi)\; =\; 0.208987\backslash ldots$.{{Cite OEIS|1=A097348|2=Decimal expansion of arccsch(2)/log(10)|mode=cs2}}

Since F_n is asymptotic to $\backslash varphi^n/\backslash sqrt5$, the number of digits in {{math|F_n}} is asymptotic to $n\backslash log\_\{10\}\backslash varphi\backslash approx\; 0.2090\backslash ,\; n$. 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 base {{mvar|b}} representation, the number of digits in {{math|F_n}} is asymptotic to $n\backslash log\_b\backslash varphi\; =\; \backslash frac\{n\; \backslash log\; \backslash varphi\}\{\backslash log\; b\}.$

Johannes Kepler observed that the ratio of consecutive Fibonacci numbers 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 $\backslash varphi\backslash colon$ {{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}}{{Citation | title = Strena seu de Nive Sexangula | year = 1611}}

$$\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{F\_\{n+1\}\}\{F\_n\}=\backslash varphi.$$

This convergence holds regardless of the starting values $U\_0$ and $U\_1$, unless $U\_1\; =\; -U\_0/\backslash varphi$. This can be verified using 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, ... . The ratio of consecutive elements in this sequence shows the same convergence towards the golden ratio.

In general, $\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{F\_\{n+m\}\}\{F\_n\}=\backslash varphi^m$

, because the ratios between consecutive Fibonacci numbers approaches $\backslash varphi$.

: File:Fibonacci tiling of the plane and approximation to Golden Ratio.gif

{{Clear}}

Since the golden ratio satisfies the equation

$$\backslash varphi^2\; =\; \backslash varphi\; +\; 1,$$

this expression can be used to decompose higher powers $\backslash varphi^n$ as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of $\backslash varphi$ and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients:

$$\backslash varphi^n\; =\; F\_n\backslash varphi\; +\; F\_\{n-1\}.$$

This equation can be proved by induction on {{math|n ≥ 1}}:

$$\backslash begin\{align\}$$

\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}

For $\backslash psi\; =\; -1/\backslash varphi$, it is also the case that $\backslash psi^2\; =\; \backslash psi\; +\; 1$ and it is also the case that

$$\backslash psi^n\; =\; F\_n\backslash psi\; +\; F\_\{n-1\}.$$

These expressions are also true for {{math|n < 1}} if the Fibonacci sequence F_n is extended to negative integers using the Fibonacci rule $F\_n\; =\; F\_\{n+2\}\; -\; F\_\{n+1\}.$

Binet's formula provides a proof that a positive integer {{mvar|x}} is a Fibonacci number if and only if at least one of $5x^2+4$ or $5x^2-4$ is a perfect square.{{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 = April 11, 2012 }} This is because Binet's formula, which can be written as $F\_n\; =\; (\backslash varphi^n\; -\; (-1)^n\; \backslash varphi^\{-n\})\; /\; \backslash sqrt\{5\}$, can be multiplied by $\backslash sqrt\{5\}\; \backslash varphi^n$ and solved as a quadratic equation in $\backslash varphi^n$ via the quadratic formula:

$$\backslash varphi^n\; =\; \backslash frac\{F\_n\backslash sqrt\{5\}\; \backslash pm\; \backslash sqrt\{5\{F\_n\}^2\; +\; 4(-1)^n\}\}\{2\}.$$

Comparing this to $\backslash varphi^n\; =\; F\_n\; \backslash varphi\; +\; F\_\{n-1\}\; =\; (F\_n\backslash sqrt\{5\}\; +\; F\_n\; +\; 2\; F\_\{n-1\})/2$, it follows that

: $$5\{F\_n\}^2\; +\; 4(-1)^n\; =\; (F\_n\; +\; 2F\_\{n-1\})^2\backslash ,.$$

In particular, the left-hand side is a perfect square.

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

\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}

alternatively denoted

$$\backslash vec\; F\_\{k+1\}\; =\; \backslash mathbf\{A\}\; \backslash vec\; F\_\{k\},$$

which yields $\backslash vec\; F\_n\; =\; \backslash mathbf\{A\}^n\; \backslash vec\; F\_0$. The eigenvalues of the matrix {{math|A}} are $\backslash varphi=\backslash tfrac12\backslash bigl(1+\backslash sqrt5~\backslash !\backslash bigr)$ and $\backslash psi=-\backslash varphi^\{-1\}=\backslash tfrac12\backslash bigl(1-\backslash sqrt5~\backslash !\backslash bigr)$ corresponding to the respective eigenvectors

$$\backslash vec\; \backslash mu=\backslash begin\{pmatrix\}\; \backslash varphi\; \backslash \backslash \; 1\; \backslash end\{pmatrix\},\; \backslash quad\; \backslash vec\backslash nu=\backslash begin\{pmatrix\}\; -\backslash varphi^\{-1\}\; \backslash \backslash \; 1\; \backslash end\{pmatrix\}.$$

As the initial value is

$$\backslash vec\; F\_0=\backslash begin\{pmatrix\}\; 1\; \backslash \backslash \; 0\; \backslash end\{pmatrix\}=\backslash frac\{1\}\{\backslash sqrt\{5\}\}\backslash vec\{\backslash mu\}\backslash ,-\backslash ,\backslash frac\{1\}\{\backslash sqrt\{5\}\}\backslash vec\{\backslash nu\},$$

it follows that the {{mvar|n}}th element is

$$\backslash begin\{align\}$$

\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}

From this, the {{mvar|n}}th element in the Fibonacci series may be read off directly as a closed-form expression:

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}.

Equivalently, the same computation may be performed by diagonalization of {{math|A}} through use of its eigendecomposition:

A^n & = S\Lambda^n S^{-1},

\end{align}

where

\Lambda=\begin{pmatrix} \varphi & 0 \\ 0 & -\varphi^{-1}\! \end{pmatrix}, \quad

S=\begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}.

The closed-form expression for the {{mvar|n}}th element in the Fibonacci series is therefore given by

& = 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}

which again yields

$$F\_n\; =\; \backslash cfrac\{\backslash varphi^n-(-\backslash varphi)^\{-n\}\}\{\backslash sqrt\{5\}\}.$$

The matrix {{math|A}} has a determinant of −1, and thus it is a 2 × 2 unimodular matrix.

This property can be understood in terms of the continued fraction representation for the golden ratio {{mvar|φ}}:

$$\backslash varphi\; =\; 1\; +\; \backslash cfrac\{1\}\{1\; +\; \backslash cfrac\{1\}\{1\; +\; \backslash cfrac\{1\}\{1\; +\; \backslash ddots\}\}\}.$$

The convergents of the continued fraction for {{mvar|φ}} are ratios of successive Fibonacci numbers: {{math|1=φ_n = F_n+1 / F_n}} is the {{mvar|n}}-th convergent, and the {{math|(n + 1)}}-st convergent can be found from the recurrence relation {{math|1=φ_n+1 = 1 + 1 / φ_n}}.{{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}} 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:

For a given {{mvar|n}}, this matrix can be computed in {{math|O(log n)}} arithmetic operations, using the exponentiation by squaring method.

Taking the determinant of both sides of this equation yields Cassini's identity,

$$(-1)^n\; =\; F\_\{n+1\}F\_\{n-1\}\; -\; \{F\_n\}^2.$$

Moreover, since {{math|AⁿA^m {{=}} A^n+m}} for any square matrix {{math|A}}, the following 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}}),

$$\backslash begin\{align\}$$

{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}

In particular, with {{math|1=m = n}},

$$\backslash begin\{align\}$$

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}

These last two identities provide a way to compute Fibonacci numbers 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).{{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}}

Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that $F\_n$ can be interpreted as the number of (possibly empty) sequences of 1s and 2s whose sum is $n-1$. This can be taken as the definition of $F\_n$ with the conventions $F\_0\; =\; 0$, meaning no such sequence exists whose sum is −1, and $F\_1\; =\; 1$, meaning the empty sequence "adds up" to 0. In the following, $|\{...\}|$ is the cardinality of a set:

: $F\_0\; =\; 0\; =\; |\backslash \{\backslash \}|$

: $F\_1\; =\; 1\; =\; |\backslash \{()\backslash \}|$

: $F\_2\; =\; 1\; =\; |\backslash \{(1)\backslash \}|$

: $F\_3\; =\; 2\; =\; |\backslash \{(1,1),(2)\backslash \}|$

: $F\_4\; =\; 3\; =\; |\backslash \{(1,1,1),(1,2),(2,1)\backslash \}|$

: $F\_5\; =\; 5\; =\; |\backslash \{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\backslash \}|$

In this manner the recurrence relation

$$F\_n\; =\; F\_\{n-1\}\; +\; F\_\{n-2\}$$

may be understood by dividing the $F\_n$ sequences into two non-overlapping sets where all sequences either begin with 1 or 2:

$$F\_n\; =\; |\backslash \{(1,...),(1,...),...\backslash \}|\; +\; |\backslash \{(2,...),(2,...),...\backslash \}|$$

Excluding the first element, the remaining terms in each sequence sum to $n-2$ or $n-3$ and the cardinality of each set is $F\_\{n-1\}$ or $F\_\{n-2\}$ giving a total of $F\_\{n-1\}+F\_\{n-2\}$ sequences, showing this is equal to $F\_n$.

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 1.{{Sfn | Lucas | 1891 | p = 4}} In symbols:

$$\backslash sum\_\{i=1\}^n\; F\_i\; =\; F\_\{n+2\}\; -\; 1$$

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

Following the same logic as before, by summing the cardinality of each set we see that

: $F\_\{n+2\}\; =\; F\_n\; +\; F\_\{n-1\}\; +\; ...\; +\; |\backslash \{(1,1,...,1,2)\backslash \}|\; +\; |\backslash \{(1,1,...,1)\backslash \}|$

... where the last two terms have the value $F\_1\; =\; 1$. From this it follows that $\backslash sum\_\{i=1\}^n\; F\_i\; =\; F\_\{n+2\}-1$.

A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:

$$\backslash sum\_\{i=0\}^\{n-1\}\; F\_\{2\; i+1\}\; =\; F\_\{2\; n\}$$

and

$$\backslash sum\_\{i=1\}^\{n\}\; F\_\{2\; i\}\; =\; F\_\{2\; n+1\}-1.$$

In words, the sum of the first Fibonacci numbers with odd index up to $F\_\{2\; n-1\}$ is the {{math|(2n)}}-th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to $F\_\{2\; n\}$ is the {{math|(2n + 1)}}-th Fibonacci number minus 1.{{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}}

A different trick may be used to prove

$$\backslash sum\_\{i=1\}^n\; F\_i^2\; =\; F\_n\; F\_\{n+1\}$$

or in words, the sum of the squares of the first Fibonacci numbers up to $F\_n$ 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 $F\_n\; \backslash times\; F\_\{n+1\}$ and decompose it into squares of size $F\_n,\; F\_\{n-1\},\; ...,\; F\_1$; from this the identity follows by comparing areas:

File:Fibonacci Squares.svg

The sequence $(F\_n)\_\{n\backslash in\backslash mathbb\; N\}$ is also considered using the symbolic method.{{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}} More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is $\backslash operatorname\{Seq\}(\backslash mathcal\{Z+Z^2\})$. Indeed, as stated above, the $n$-th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of $n-1$ using terms 1 and 2.

It follows that the ordinary generating function of the Fibonacci sequence, $\backslash sum\_\{i=0\}^\backslash infty\; F\_iz^i$, is the rational function $\backslash frac\{z\}\{1-z-z^2\}.$

Fibonacci identities often can be easily proved using mathematical induction.

For example, reconsider

$$\backslash sum\_\{i=1\}^n\; F\_i\; =\; F\_\{n+2\}\; -\; 1.$$

Adding $F\_\{n+1\}$ to both sides gives

: $\backslash sum\_\{i=1\}^n\; F\_i\; +\; F\_\{n+1\}\; =\; F\_\{n+1\}\; +\; F\_\{n+2\}\; -\; 1$

and so we have the formula for $n+1$

$$\backslash sum\_\{i=1\}^\{n+1\}\; F\_i\; =\; F\_\{n+3\}\; -\; 1$$

Similarly, add $\{F\_\{n+1\}\}^2$ to both sides of

$$\backslash sum\_\{i=1\}^n\; F\_i^2\; =\; F\_n\; F\_\{n+1\}$$

to give

$$\backslash sum\_\{i=1\}^n\; F\_i^2\; +\; \{F\_\{n+1\}\}^2\; =\; F\_\{n+1\}\backslash left(F\_n\; +\; F\_\{n+1\}\backslash right)$$

$$\backslash sum\_\{i=1\}^\{n+1\}\; F\_i^2\; =\; F\_\{n+1\}F\_\{n+2\}$$

The Binet formula is

$$\backslash sqrt5F\_n\; =\; \backslash varphi^n\; -\; \backslash psi^n.$$

This can be used to prove Fibonacci identities.

For example, to prove that $\backslash sum\_\{i=1\}^n\; F\_i\; =\; F\_\{n+2\}\; -\; 1$

note that the left hand side multiplied by $\backslash sqrt5$ becomes

\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}

as required, using the facts $\backslash varphi\backslash psi\; =-\; 1$ and $\backslash varphi-\backslash psi=\backslash sqrt5$ to simplify the equations.

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

{{Main|Cassini and Catalan identities}}

Cassini's identity states that

$$\{F\_n\}^2\; -\; F\_\{n+1\}F\_\{n-1\}\; =\; (-1)^\{n-1\}$$

Catalan's identity is a generalization:

$$\{F\_n\}^2\; -\; F\_\{n+r\}F\_\{n-r\}\; =\; (-1)^\{n-r\}\{F\_r\}^2$$

$$F\_m\; F\_\{n+1\}\; -\; F\_\{m+1\}\; F\_n\; =\; (-1)^n\; F\_\{m-n\}$$

$$F\_\{2\; n\}\; =\; \{F\_\{n+1\}\}^2\; -\; \{F\_\{n-1\}\}^2\; =\; F\_n\; \backslash left\; (F\_\{n+1\}+F\_\{n-1\}\; \backslash right\; )\; =\; F\_nL\_n$$

where {{math|L_n}} is the {{mvar|n}}-th Lucas number. The last is an identity for doubling {{mvar|n}}; other identities of this type are

$$F\_\{3\; n\}\; =\; 2\{F\_n\}^3\; +\; 3\; F\_n\; F\_\{n+1\}\; F\_\{n-1\}\; =\; 5\{F\_n\}^3\; +\; 3\; (-1)^n\; F\_n$$

by Cassini's identity.

$$F\_\{3\; n+1\}\; =\; \{F\_\{n+1\}\}^3\; +\; 3\; F\_\{n+1\}\{F\_n\}^2\; -\; \{F\_n\}^3$$

$$F\_\{3\; n+2\}\; =\; \{F\_\{n+1\}\}^3\; +\; 3\; \{F\_\{n+1\}\}^2\; F\_n\; +\; \{F\_n\}^3$$

$$F\_\{4\; n\}\; =\; 4\; F\_n\; F\_\{n+1\}\; \backslash left\; (\{F\_\{n+1\}\}^2\; +\; 2\{F\_n\}^2\; \backslash right\; )\; -\; 3\{F\_n\}^2\; \backslash left\; (\{F\_n\}^2\; +\; 2\{F\_\{n+1\}\}^2\; \backslash right\; )$$

These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number.

More generally,

$$F\_\{k\; n+c\}\; =\; \backslash sum\_\{i=0\}^k\; \backslash binom\; k\; i\; F\_\{c-i\}\; \{F\_n\}^i\; \{F\_\{n+1\}\}^\{k-i\}.$$

or alternatively

$$F\_\{k\; n+c\}\; =\; \backslash sum\_\{i=0\}^k\; \backslash binom\; k\; i\; F\_\{c+i\}\; \{F\_n\}^i\; \{F\_\{n-1\}\}^\{k-i\}.$$

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

The generating function of the Fibonacci sequence is the power series

s(z) = \sum_{k=0}^\infty F_k z^k = 0 + z + z^2 + 2z^3 + 3z^4 + 5z^5 + \dots.

This series is convergent for any complex number $z$ satisfying and its sum has a simple closed form:{{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 }}

$$s(z)=\backslash frac\{z\}\{1-z-z^2\}.$$

This can be proved by multiplying by $(1-z-z^2)$:

$$\backslash begin\{align\}$$

(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}

where all terms involving $z^k$ for $k\; \backslash ge\; 2$ cancel out because of the defining Fibonacci recurrence relation.

The partial fraction decomposition is given by

$$s(z)\; =\; \backslash frac\{1\}\{\backslash sqrt5\}\backslash left(\backslash frac\{1\}\{1\; -\; \backslash varphi\; z\}\; -\; \backslash frac\{1\}\{1\; -\; \backslash psi\; z\}\backslash right)$$

where $\backslash varphi\; =\; \backslash tfrac12\backslash left(1\; +\; \backslash sqrt\{5\}\backslash right)$ is the golden ratio and $\backslash psi\; =\; \backslash tfrac12\backslash left(1\; -\; \backslash sqrt\{5\}\backslash right)$ is its conjugate.

The related function $z\; \backslash mapsto\; -s\backslash left(-1/z\backslash right)$ is the generating function for the negafibonacci numbers, and $s(z)$ satisfies the functional equation

$$s(z)\; =\; s\backslash !\backslash left(-\backslash frac\{1\}\{z\}\backslash right).$$

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

Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as

$$\backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{F\_\{2\; k-1\}\}\; =\; \backslash frac\{\backslash sqrt\{5\}\}\{4\}\; \backslash ;\; \backslash vartheta\_2\backslash !\backslash left(0,\; \backslash frac\{3-\backslash sqrt\; 5\}\{2\}\backslash right)^2\; ,$$

and the sum of squared reciprocal Fibonacci numbers as

$$\backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{\{F\_k\}^2\}\; =\; \backslash frac\{5\}\{24\}\; \backslash !\backslash left(\backslash vartheta\_2\backslash !\backslash left(0,\; \backslash frac\{3-\backslash sqrt\; 5\}\{2\}\backslash right)^4\; -\; \backslash vartheta\_4\backslash !\backslash left(0,\; \backslash frac\{3-\backslash sqrt\; 5\}\{2\}\backslash right)^4\; +\; 1\; \backslash right).$$

If we add 1 to each Fibonacci number in the first sum, there is also the closed form

$$\backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{1+F\_\{2\; k-1\}\}\; =\; \backslash frac\{\backslash sqrt\{5\}\}\{2\},$$

and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,

$$\backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{(-1)^\{k+1\}\}\{\backslash sum\_\{j=1\}^k\; \{F\_\{j\}\}^2\}\; =\; \backslash frac\{\backslash sqrt\{5\}-1\}\{2\}\; .$$

The sum of all even-indexed reciprocal Fibonacci numbers isLandau (1899) quoted according Borwein, Page 95, Exercise 3b.

$$\backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{F\_\{2\; k\}\}\; =\; \backslash sqrt\{5\}\; \backslash left(L(\backslash psi^2)\; -\; L(\backslash psi^4)\backslash right)$$

with the Lambert series $\backslash textstyle\; L(q)\; :=\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\{q^k\}\{1-q^k\}\; ,$ since $\backslash textstyle\; \backslash frac\{1\}\{F\_\{2\; k\}\}\; =\; \backslash sqrt\{5\}\; \backslash left(\backslash frac\{\backslash psi^\{2\; k\}\}\{1-\backslash psi^\{2\; k\}\}\; -\; \backslash frac\{\backslash psi^\{4\; k\}\}\{1-\backslash psi^\{4\; k\}\}\; \backslash right)\backslash !.$

So the reciprocal Fibonacci constant is{{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}}

$$\backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{F\_k\}\; =\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{F\_\{2\; k-1\}\}\; +\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\; \{1\}\{F\_\{2\; k\}\}\; =\; 3.359885666243\; \backslash dots$$

Moreover, this number has been proved irrational by Richard André-Jeannin.{{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}}

Millin's series gives the identity{{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}}

$$\backslash sum\_\{k=0\}^\{\backslash infty\}\; \backslash frac\{1\}\{F\_\{2^k\}\}\; =\; \backslash frac\{7\; -\; \backslash sqrt\{5\}\}\{2\},$$

which follows from the closed form for its partial sums as {{mvar|N}} tends to infinity:

$$\backslash sum\_\{k=0\}^N\; \backslash frac\{1\}\{F\_\{2^k\}\}\; =\; 3\; -\; \backslash frac\{F\_\{2^N-1\}\}\{F\_\{2^N\}\}.$$

Every third number of the sequence is even (a multiple of $F\_3=2$) and, more generally, every {{mvar|k}}-th number of the sequence is a multiple of F_k. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property{{Citation | first = Paulo | last = Ribenboim | author-link = Paulo Ribenboim | title = My Numbers, My Friends | publisher = Springer-Verlag | year = 2000}}{{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 }}

$$\backslash gcd(F\_a,F\_b,F\_c,\backslash ldots)\; =\; F\_\{\backslash gcd(a,b,c,\backslash ldots)\}\backslash ,$$

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 because both $F\_1=1$ and $F\_2\; =\; 1$. That is,

: $\backslash gcd(F\_n,\; F\_\{n+1\})\; =\; \backslash gcd(F\_n,\; F\_\{n+2\})\; =\; \backslash gcd(F\_\{n+1\},\; F\_\{n+2\})\; =\; 1$

for every {{mvar|n}}.

Every prime number {{mvar|p}} divides a Fibonacci number that can be determined by the value of {{mvar|p}} modulo 5. If {{mvar|p}} is congruent to 1 or 4 modulo 5, then {{mvar|p}} divides {{math|F_p−1}}, and if {{mvar|p}} is congruent to 2 or 3 modulo 5, then, {{mvar|p}} divides {{math|F_p+1}}. The remaining case is that {{math|1=p = 5}}, and in this case {{mvar|p}} divides F_p.

These cases can be combined into a single, non-piecewise formula, using the Legendre symbol:{{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 $F\_\{p-\backslash varepsilon\}/p$

| volume = 25

| year = 1982| hdl = 10338.dmlcz/137492

| hdl-access = free

}}. Williams calls this property "well known".

$$p\; \backslash mid\; F\_\{p\; \backslash ;-\backslash ,\; \backslash left(\backslash frac\{5\}\{p\}\backslash right)\}.$$

The above formula can be used as a primality test in the sense that if

$$n\; \backslash mid\; F\_\{n\; \backslash ;-\backslash ,\; \backslash left(\backslash frac\{5\}\{n\}\backslash right)\},$$

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 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_m (mod n)}} efficiently using the matrix form. Thus

Here the matrix power {{math|A^m}} is calculated using modular exponentiation, which can be adapted to matrices.Prime Numbers, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.

{{Main|Fibonacci prime}}

A Fibonacci prime is a Fibonacci number that is prime. The first few are:{{Cite OEIS|1=A005478|2=Prime Fibonacci numbers|mode=cs2}}

: 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.{{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

}}

{{math|F_kn}} is divisible by {{math|F_n}}, so, apart from {{math|1=F₄ = 3}}, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

No Fibonacci number greater than {{math|1=F₆ = 8}} is one greater or one less than a prime number.{{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}}

The only nontrivial square Fibonacci number is 144.{{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}} Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.{{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}} In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.{{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}}

1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.{{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 }}

No Fibonacci number can be a perfect number.{{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 }} More generally, no Fibonacci number other than 1 can be multiply perfect,{{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 }} and no ratio of two Fibonacci numbers can be perfect.{{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 }}

With the exceptions of 1, 8 and 144 ({{math|1=F₁ = F₂}}, {{math|F₆}} and {{math|F₁₂}}) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).{{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}} As a result, 8 and 144 ({{math|F₆}} and {{math|F₁₂}}) are the only Fibonacci numbers that are the product of other Fibonacci numbers.{{Cite OEIS|1=A235383|2=Fibonacci numbers that are the product of other Fibonacci numbers|mode=cs2}}

The divisibility of Fibonacci numbers by a prime {{mvar|p}} is related to the Legendre symbol $\backslash bigl(\backslash tfrac\{p\}\{5\}\backslash bigr)$ which is evaluated as follows:

If {{mvar|p}} is a prime number then

$$F\_p\; \backslash equiv\; \backslash left(\backslash frac\{p\}\{5\}\backslash right)\; \backslash pmod\; p\; \backslash quad\; \backslash text\{and\}\backslash quad\; F\_\{p-\backslash left(\backslash frac\{p\}\{5\}\backslash right)\}\; \backslash equiv\; 0\; \backslash pmod\; p.$${{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}}{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.25–28 | pp = 73–74}}

For example,

$$\backslash begin\{align\}$$

\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}

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

$$F\_\{p-\backslash left(\backslash frac\{p\}\{5\}\backslash right)\}\; \backslash equiv\; 0\; \backslash pmod\{p^2\}.$$

Such primes (if there are any) would be called Wall–Sun–Sun primes.

Also, if {{math|p ≠ 5}} is an odd prime number then:{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.28 | pp = 73–74}}

$$5\; \{F\_\{\backslash frac\{p\; \backslash pm\; 1\}\{2\}\}\}^2\; \backslash equiv\; \backslash begin\{cases\}$$

\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}

Example 1. {{math|1=p = 7}}, in this case {{math|1=p ≡ 3 (mod 4)}} and we have:

$$\backslash bigl(\backslash tfrac\{7\}\{5\}\backslash bigr)\; =\; -1:\; \backslash qquad\; \backslash tfrac\{1\}\{2\}\backslash left(5\; \backslash bigl(\backslash tfrac\{7\}\{5\}\backslash bigr)+3\; \backslash right\; )\; =-1,\; \backslash quad\; \backslash tfrac\{1\}\{2\}\; \backslash left(5\backslash bigl(\backslash tfrac\{7\}\{5\}\backslash bigr)-3\; \backslash right\; )=-4.$$

$$F\_3=2\; \backslash text\{\; and\; \}\; F\_4=3.$$

$$5\{F\_3\}^2=20\backslash equiv\; -1\; \backslash pmod\; \{7\}\backslash ;\backslash ;\backslash text\{\; and\; \}\backslash ;\backslash ;5\{F\_4\}^2=45\backslash equiv\; -4\; \backslash pmod\; \{7\}$$

Example 2. {{math|1=p = 11}}, in this case {{math|1=p ≡ 3 (mod 4)}} and we have:

$$\backslash bigl(\backslash tfrac\{11\}\{5\}\backslash bigr)\; =\; +1:\; \backslash qquad\; \backslash tfrac\{1\}\{2\}\backslash left(\; 5\backslash bigl(\backslash tfrac\{11\}\{5\}\backslash bigr)+3\; \backslash right)=4,\; \backslash quad\; \backslash tfrac\{1\}\{2\}\; \backslash left(5\backslash bigl(\backslash tfrac\{11\}\{5\}\backslash bigr)-\; 3\; \backslash right)=1.$$

$$F\_5=5\; \backslash text\{\; and\; \}\; F\_6=8.$$

$$5\{F\_5\}^2=125\backslash equiv\; 4\; \backslash pmod\; \{11\}\; \backslash ;\backslash ;\backslash text\{\; and\; \}\backslash ;\backslash ;5\{F\_6\}^2=320\backslash equiv\; 1\; \backslash pmod\; \{11\}$$

Example 3. {{math|1=p = 13}}, in this case {{math|1=p ≡ 1 (mod 4)}} and we have:

$$\backslash bigl(\backslash tfrac\{13\}\{5\}\backslash bigr)\; =\; -1:\; \backslash qquad\; \backslash tfrac\{1\}\{2\}\backslash left(5\backslash bigl(\backslash tfrac\{13\}\{5\}\backslash bigr)-5\; \backslash right)\; =-5,\; \backslash quad\; \backslash tfrac\{1\}\{2\}\backslash left(5\backslash bigl(\backslash tfrac\{13\}\{5\}\backslash bigr)+\; 5\; \backslash right)=0.$$

$$F\_6=8\; \backslash text\{\; and\; \}\; F\_7=13.$$

$$5\{F\_6\}^2=320\backslash equiv\; -5\; \backslash pmod\; \{13\}\; \backslash ;\backslash ;\backslash text\{\; and\; \}\backslash ;\backslash ;5\{F\_7\}^2=845\backslash equiv\; 0\; \backslash pmod\; \{13\}$$

Example 4. {{math|1=p = 29}}, in this case {{math|1=p ≡ 1 (mod 4)}} and we have:

$$\backslash bigl(\backslash tfrac\{29\}\{5\}\backslash bigr)\; =\; +1:\; \backslash qquad\; \backslash tfrac\{1\}\{2\}\backslash left(5\backslash bigl(\backslash tfrac\{29\}\{5\}\backslash bigr)-5\; \backslash right)=0,\; \backslash quad\; \backslash tfrac\{1\}\{2\}\backslash left(5\backslash bigl(\backslash tfrac\{29\}\{5\}\backslash bigr)+5\; \backslash right)=5.$$

$$F\_\{14\}=377\; \backslash text\{\; and\; \}\; F\_\{15\}=610.$$

$$5\{F\_\{14\}\}^2=710645\backslash equiv\; 0\; \backslash pmod\; \{29\}\; \backslash ;\backslash ;\backslash text\{\; and\; \}\backslash ;\backslash ;5\{F\_\{15\}\}^2=1860500\backslash equiv\; 5\; \backslash pmod\; \{29\}$$

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

For example,

$$F\_1\; =\; 1,\backslash \; F\_3\; =\; 2,\backslash \; F\_5\; =\; 5,\backslash \; F\_7\; =\; 13,\backslash \; F\_9\; =\; \{\backslash color\{Red\}34\}\; =\; 2\; \backslash cdot\; 17,\backslash \; F\_\{11\}\; =\; 89,\backslash \; F\_\{13\}\; =\; 233,\backslash \; F\_\{15\}\; =\; \{\backslash color\{Red\}610\}\; =\; 2\; \backslash cdot\; 5\; \backslash cdot\; 61.$$

All known factors of Fibonacci numbers {{math|F(i)}} for all {{math|i < 50000}} are collected at the relevant repositories.{{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}}.{{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}}.

{{Main|Pisano period}}

If the members of the Fibonacci sequence are taken mod {{mvar|n}}, the resulting sequence is periodic with period at most {{math|6n}}.{{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 }} The lengths of the periods for various {{mvar|n}} form the so-called Pisano periods.{{Cite OEIS|1=A001175|2=Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n|mode=cs2}} 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 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.

{{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 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 numbers using a modification of Binet's formula.
• Starting with other integers. Lucas numbers have {{math|1=L₁ = 1}}, {{math|1=L₂ = 3}}, and {{math|1=L_n = L_n−1 + L_n−2}}. Primefree sequences 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 numbers have {{math|1=P_n = 2P_n−1 + P_n−2}}. 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 numbers 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.{{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}}

File:Pascal triangle fibonacci.svg.]]

The Fibonacci numbers occur as the sums of binomial coefficients in the "shallow" diagonals of Pascal's triangle:{{Sfn | Lucas | 1891 | p = 7}}

$$F\_n\; =\; \backslash sum\_\{k=0\}^\{\backslash left\backslash lfloor\backslash frac\{n-1\}\{2\}\backslash right\backslash rfloor\}\; \backslash binom\{n-k-1\}\{k\}.$$

This can be proved by expanding the generating function

$$\backslash frac\{x\}\{1-x-x^2\}\; =\; x\; +\; x^2(1+x)\; +\; x^3(1+x)^2\; +\; \backslash dots\; +\; x^\{k+1\}(1+x)^k\; +\; \backslash dots\; =\; \backslash sum\backslash limits\_\{n=0\}^\backslash infty\; F\_n\; x^n$$

and collecting like terms of $x^n$.

To see how the formula is used, we can arrange the sums by the number of terms present:

:

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

File:Fibonacci climbing stairs.svg

These numbers also give the solution to certain enumerative problems,{{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}} 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 compositions); there are {{math|F_n+1}} ways to do this (equivalently, it's also the number of domino tilings of the $2\backslash times\; n$ rectangle). For example, there are {{math|1=F₅₊₁ = F₆ = 8}} ways one can climb a staircase of 5 steps, taking one or two steps at a time:

:

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 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 strings, or equivalently, among the subsets of a given set.

• The number of binary strings of length {{mvar|n}} without consecutive {{math|1}}s is the Fibonacci number {{math|F_n+2}}. For example, out of the 16 binary strings of length 4, there are {{math|1=F₆ = 8}} without consecutive {{math|1}}s—they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of Fibbinary numbers. Equivalently, {{math|F_n+2}} 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 0 and 2 with 10, and drop the last zero.
• The number of binary strings of length {{mvar|n}} without an odd number of consecutive {{math|1}}s is the Fibonacci number {{math|F_n+1}}. For example, out of the 16 binary strings of length 4, there are {{math|1=F₅ = 5}} without an odd number of consecutive {{math|1}}s—they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets {{mvar|S}} of {{math|{{mset|1, ..., n}}}} without an odd number of consecutive integers is {{math|F_n+1}}. A bijection with the sums to {{mvar|n}} is to replace 1 with 0 and 2 with 11.
• The number of binary strings of length {{mvar|n}} without an even number of consecutive {{math|0}}s or {{math|1}}s is {{math|2F_n}}. For example, out of the 16 binary strings of length 4, there are {{math|1=2F₄ = 6}} without an even number of consecutive {{math|0}}s or {{math|1}}s—they are 0001, 0111, 0101, 1000, 1010, 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 his solving Hilbert's tenth problem.{{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}}
• 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 $$(F\_n\; F\_\{n+3\})^2\; +\; (2\; F\_\{n+1\}F\_\{n+2\})^2\; =\; \{F\_\{2\; n+3\}\}^2.$$ 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), ... . The middle side of each of these triangles is the sum of the three sides of the preceding triangle.{{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}}

• 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 maps.{{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.

File:Fibonacci Tree 6.svgs green; heights red.
The keys in the left spine are Fibonacci numbers.]]

• The Fibonacci numbers are important in computational run-time analysis of 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.{{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}}
• 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 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.{{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.
• Fibonacci numbers are used by some pseudorandom number generators.
• 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.{{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 }}
• The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law.{{Citation | title = Amiga ROM Kernel Reference Manual | publisher = Addison–Wesley | year = 1991}}{{Citation | url = https://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression | contribution = IFF | title = Multimedia Wiki}}
• 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.{{citation|author=Dean Leffingwell |url=https://www.scaledagileframework.com/story/ |title=Story |publisher=Scaled Agile Framework |date=2021-07-01 |accessdate=2022-08-15}}
• Fibonacci coding
• Negafibonacci coding

{{Further|Patterns in nature}}

{{see also|Golden ratio#Nature}}

File:FibonacciChamomile.PNG 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,{{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 }} such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,{{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}} the flowering of artichoke, the arrangement of a pine cone,{{Citation| first=A | last=Brousseau |title=Fibonacci Statistics in Conifers | journal=Fibonacci Quarterly |year=1969 |issue=7 |pages=525–32}} and the family tree of honeybees.{{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–}}{{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}} Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.{{sfn|Livio|2003|p=110}} Field 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 parastichies (spiral phyllotaxis) of plants were frequently expressed as fractions involving Fibonacci numbers.{{Citation |first =Franck |last = Varenne |title = Formaliser le vivant - Lois, Théories, Modèles | accessdate = 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 [...].|lang = fr}}

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

File:SunflowerModel.svg

A model for the pattern of florets in the head of a sunflower was proposed by {{ill|Helmut Vogel|de|Helmut Vogel (Physiker)}} in 1979.{{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}} This has the form

$$\backslash theta\; =\; \backslash frac\{2\backslash pi\}\{\backslash varphi^2\}\; n,\backslash \; r\; =\; c\; \backslash sqrt\{n\}$$

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( j):F( j + 1)}}, the nearest neighbors of floret number {{mvar|n}} are those at {{math|n ± F( 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.{{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 }}

Fibonacci numbers also appear in the ancestral pedigrees of bees (which are haplodiploids), according to the following rules:

• If an egg is laid but not fertilized, it produces a male (or 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_n}}, is the number of female ancestors, which is {{math|F_n−1}}, plus the number of male ancestors, which is {{math|F_n−2}}.{{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. }}Yanega, D. 1996. Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae). J. Kans. Ent. Soc. 69 Suppl.: 98-115. This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

File:X chromosome ancestral line Fibonacci sequence.svg

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.{{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}} 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 ($F\_1=1$), and at his parents' generation, his X chromosome came from a single parent {{nowrap|($F\_2=1$)}}. 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|($F\_3=2$)}}. 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|($F\_4=3$)}}. Five great-great-grandparents contributed to the male descendant's X chromosome {{nowrap|($F\_5=5$)}}, 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 population founder appears on all lines of the genealogy.)

File:BerlinVictoryColumnStairs.jpg

• In optics, when a beam of light shines at an angle through two stacked transparent plates of different materials of different refractive indexes, 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 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 register in golden ratio base {{mvar|φ}} being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.{{Citation | url = https://www.encyclopediaofmath.org/index.php/Zeckendorf_representation | contribution = Zeckendorf representation | title = Encyclopedia of Math}}
• 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.{{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

}}

• Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of economics.{{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 }} 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 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}}.

• {{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}}

{{Notelist}}

{{Reflist}}

• {{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.

| authorlink =Jonathan Borwein

| authorlink2=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 = 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}}

{{Wikiquote}}

{{Wikibooks|Fibonacci number program}}

• {{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}}

{{Classes of natural numbers}}

{{Metallic ratios}}

{{Series (mathematics)}}

{{Fibonacci}}

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{{Interwiki extra|qid=Q47577}}

Category:Articles containing proofs

Category:Integer sequences