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Fibonacci sequence

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In mathematics, the Fibonacci sequence is a 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 Template:Nowrap. 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, ... (sequence A000045 in the OEIS)
File:Fibonacci Squares.svg
A tiling with squares 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 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 {{#invoke:Lang|lang}}.Template:Sfn

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 graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, 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 expresses the Template:Mvar-th Fibonacci number in terms of Template:Mvar and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as Template:Mvar increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.

DefinitionEdit

File:Fibonacci Spiral.svg
The Fibonacci spiral: an approximation of the golden spiral 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 relationTemplate:Sfn <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 Template:Math.

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 Template:Math.Template:SfnTemplate:Sfn

The first 20 Fibonacci numbers Template:Math are:

Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181

HistoryEdit

IndiaEdit

Template:See also

File:Fibonacci Sanskrit prosody.svg
Thirteen (Template:Math) ways of arranging long and short syllables in a cadence of length six. Eight (Template:Math) end with a short syllable and five (Template:Math) end with a long syllable.

The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.<ref name="HistoriaMathematica">Template:Citation</ref><ref name="knuth-v1">Template:Citation</ref>Template:Sfn 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 Template:Mvar units is Template:Math.<ref name="Donald Knuth 2006 50">Template:Citation</ref>

Knowledge of the Fibonacci sequence was expressed as early as Pingala (Template: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 Template:Mvar beats (Template:Math) is obtained by adding one [S] to the Template:Math cases and one [L] to the Template:Math cases.<ref>Template:Citation</ref> Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).<ref name="HistoriaMathematica"/><ref name=GlobalScience>Template:Citation</ref> 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):Template:Sfn

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].Template:Efn

Hemachandra (c. 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 ... of the next mātrā-vṛtta."Template:Sfn<ref>Template:Citation</ref>

EuropeEdit

File:Liber abbaci magliab f124r.jpg
lang}} 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 {{#invoke:Lang|lang}} (The Book of Calculation, 1202) by Fibonacci,Template:Sfn<ref>Template:Citation</ref> where it is used to calculate the growth of rabbit populations.<ref>Template:Citation</ref> 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 Template:Mvar-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month Template:Math) plus the number of pairs alive last month (month Template:Math). The number in the Template:Mvar-th month is the Template:Mvar-th Fibonacci number.<ref>Template:Citation</ref>

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.<ref>Template:Citation</ref>

File:Fibonacci Rabbits.svg
Solution to Fibonacci rabbit problem: 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 Fn.

Relation to the golden ratioEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Closed-form expression Edit

Like every sequence defined by a homogeneous linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression.<ref>Template:Cite book 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>Template:Citation</ref>

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

where

<math display=block> \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\ldots </math>

is the golden ratio, and <math>\psi</math> is its conjugate:Template:Sfn

<math display=block> \psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = - {1 \over \varphi} \approx -0.61803\,39887\ldots. </math>

Since <math>\psi = -\varphi^{-1}</math>, this 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,Template:Sfn note that <math>\varphi</math> and <math>\psi</math> are both solutions of the equation <math display=inline>x^2 = x + 1</math> and thus <math>x^n = x^{n-1} + x^{n-2},</math> so the powers of <math>\varphi</math> and <math>\psi</math> satisfy the Fibonacci recursion. 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 Template:Mvar and Template:Mvar, the sequence defined by

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

satisfies the same recurrence,

<math display=block>\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}</math>

If Template:Mvar and Template:Mvar are chosen so that Template:Math and Template:Math then the resulting sequence Template:Math must be the Fibonacci sequence. This is the same as requiring Template:Mvar and Template:Mvar satisfy the system of equations:

<math display=block> \left\{\begin{align} a + b &= 0 \\ \varphi a + \psi b &= 1\end{align}\right. </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 Template:Math and Template:Math to be arbitrary constants, a more general solution is:

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

where

<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>

Computation by roundingEdit

Since <math display=inline>\left|\frac{\psi^{n}}{\sqrt 5}\right| < \frac{1}{2}</math> for all Template:Math, the number Template:Math 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 Template:Mvar grows, being less than 0.1 for Template:Math, and less than 0.01 for Template:Math. This formula is easily inverted to find an index of a Fibonacci number Template:Mvar: <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 Template:Mvar: <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>Template:Cite OEIS</ref> and <math>\log_{10}(\varphi) = 0.208987\ldots</math>.<ref>Template:Cite OEIS</ref>

MagnitudeEdit

Since Fn is asymptotic to <math>\varphi^n/\sqrt5</math>, the number of digits in Template:Math is asymptotic to <math>n\log_{10}\varphi\approx 0.2090\, n</math>. As a consequence, for every integer Template:Math there are either 4 or 5 Fibonacci numbers with Template:Mvar decimal digits.

More generally, in the base Template:Mvar representation, the number of digits in Template:Math is asymptotic to <math>n\log_b\varphi = \frac{n \log \varphi}{\log b}.</math>

Limit of consecutive quotientsEdit

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 <math>\varphi\colon </math> <ref>Template:Citation</ref><ref>Template:Citation</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. 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, <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
Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous

Decomposition of powersEdit

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 relationships yield Fibonacci numbers as the linear coefficients: <math display=block>\varphi^n = F_n\varphi + F_{n-1}.</math> This equation can be proved by induction on Template:Math: <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 Template:Math if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule <math>F_n = F_{n+2} - F_{n+1}.</math>

IdentificationEdit

Binet's formula provides a proof that a positive integer Template:Mvar 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 perfect square.<ref>Template:Citation</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 formEdit

A 2-dimensional system of linear difference equations 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 eigenvalues of the matrix Template:Math are <math>\varphi=\tfrac12\bigl(1+\sqrt5~\!\bigr)</math> and <math>\psi=-\varphi^{-1}=\tfrac12\bigl(1-\sqrt5~\!\bigr)</math> corresponding to the respective eigenvectors <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 Template:Mvarth 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 Template:Mvarth 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 diagonalization of Template:Math 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 Template:Mvarth 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 Template:Math 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 Template:Mvar:

<math display=block>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}.</math>

The convergents of the continued fraction for Template:Mvar are ratios of successive Fibonacci numbers: Template:Math is the Template:Mvar-th convergent, and the Template:Math-st convergent can be found from the recurrence relation Template:Math.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 Template:Mvar, this matrix can be computed in Template:Math arithmetic operations, 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 Template:Math for any square matrix Template:Math, 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 Template:Mvar into Template:Math), <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 Template:Math, <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 recursively in Template:Math arithmetic operations. This matches the time for computing the Template:Mvar-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>Template:Citation</ref>

Combinatorial identitiesEdit

Combinatorial proofsEdit

Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that <math>F_n</math> can be interpreted as the number of (possibly empty) sequences of 1s and 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 −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:

<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 Template:Mvar-th is equal to the Template:Math-th Fibonacci number minus 1.Template:Sfn 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 1 rather than the first 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 odd index up to <math>F_{2 n-1}</math> is the Template:Math-th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to <math>F_{2 n}</math> is the Template:Math-th Fibonacci number minus 1.<ref>Template:Citation</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 Template:Mvar-th and Template:Math-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

Symbolic methodEdit

The sequence <math>(F_n)_{n\in\mathbb N}</math> is also considered using the symbolic method.<ref>Template:Citation</ref> More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is <math>\operatorname{Seq}(\mathcal{Z+Z^2})</math>. Indeed, as stated above, the <math>n</math>-th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of <math>n-1</math> using terms 1 and 2.

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

Induction proofsEdit

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 proofsEdit

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 identitiesEdit

Numerous other identities can be derived using various methods. Here are some of them:<ref name="MathWorld">{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:FibonacciNumber%7CFibonacciNumber.html}} |title = Fibonacci Number |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>

Cassini's and Catalan's identitiesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 identityEdit

<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 Template:Math is the Template:Mvar-th Lucas number. The last is an identity for doubling Template:Mvar; 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 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 Template:Math in this formula, one gets again the formulas of the end of above section Matrix form.

Generating functionEdit

The 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 + \dots. </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>Template:Citation</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.

The related function <math display=inline>z \mapsto -s\left(-1/z\right)</math> is the generating function for the negafibonacci numbers, and <math>s(z)</math> satisfies the functional equation

<math display=block>s(z) = s\!\left(-\frac{1}{z}\right).</math>

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 sumsEdit

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 <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>Landau (1899) quoted according 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>Template:Cite OEIS</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 by Richard André-Jeannin.<ref>Template:Citation</ref>

Millin's series gives the identity<ref>Template:Citation</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 Template:Mvar 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 divisibilityEdit

Divisibility propertiesEdit

Every third number of the sequence is even (a multiple of <math>F_3=2</math>) and, more generally, every Template:Mvar-th number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property<ref>Template:Citation</ref><ref>Template:Citation</ref> <math display=block>\gcd(F_a,F_b,F_c,\ldots) = F_{\gcd(a,b,c,\ldots)}\,</math> where Template:Math 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 Template:Tmath and Template:Tmath.)

In particular, any three consecutive Fibonacci numbers are pairwise 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 Template:Mvar.

Every prime number Template:Mvar divides a Fibonacci number that can be determined by the value of Template:Mvar modulo 5. If Template:Mvar is congruent to 1 or 4 modulo 5, then Template:Mvar divides Template:Math, and if Template:Mvar is congruent to 2 or 3 modulo 5, then, Template:Mvar divides Template:Math. The remaining case is that Template:Math, and in this case Template:Mvar divides Fp.

<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>Template:Citation. Williams calls this property "well known".</ref> <math display=block>p \mid F_{p \;-\, \left(\frac{5}{p}\right)}.</math>

Primality testingEdit

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 Template:Mvar is a prime, and if it fails to hold, then Template:Mvar is definitely not a prime. If Template:Mvar is composite and satisfies the formula, then Template:Mvar is a Fibonacci pseudoprime. When Template:Mvar is largeTemplate:Sndsay a 500-bit numberTemplate:Sndthen we can calculate Template:Math 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 Template:Math is calculated using modular exponentiation, which can be adapted to matrices.<ref>Prime Numbers, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.</ref>

Fibonacci primesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

A Fibonacci prime is a Fibonacci number that is prime. The first few are:<ref>Template:Cite OEIS</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>Template:Citation</ref>

Template:Math is divisible by Template:Math, so, apart from Template:Math, 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 Template:Math is one greater or one less than a prime number.<ref>Template:Citation</ref>

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

1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.<ref>Template:Citation</ref>

No Fibonacci number can be a perfect number.<ref name="Luca2000">Template:Citation</ref> More generally, no Fibonacci number other than 1 can be multiply perfect,<ref name="BGLLHT2011">Template:Citation</ref> and no ratio of two Fibonacci numbers can be perfect.<ref name="LucaMH2010">Template:Citation</ref>

Prime divisorsEdit

With the exceptions of 1, 8 and 144 (Template:Math, Template:Math and Template:Math) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).<ref>Template:Citation</ref> As a result, 8 and 144 (Template:Math and Template:Math) are the only Fibonacci numbers that are the product of other Fibonacci numbers.<ref>Template:Cite OEIS</ref>

The divisibility of Fibonacci numbers by a prime Template:Mvar 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 Template:Mvar 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>Template:Citation</ref>Template:Sfn

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 Template:Mvar 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 primes.

Also, if Template:Math is an odd prime number then:Template:Sfn <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. Template:Math, in this case Template:Math 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. Template:Math, in this case Template:Math 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. Template:Math, in this case Template:Math 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. Template:Math, in this case Template:Math 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 Template:Mvar, all odd prime divisors of Template:Math are congruent to 1 modulo 4, implying that all odd divisors of Template:Math (as the products of odd prime divisors) are congruent to 1 modulo 4.Template:Sfn

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 Template:Math for all Template:Math are collected at the relevant repositories.<ref>Template:Citation collects all known factors of Template:Math with Template:Math.</ref><ref>Template:Citation collects all known factors of Template:Math with Template:Math.</ref>

Periodicity modulo nEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

If the members of the Fibonacci sequence are taken mod Template:Mvar, the resulting sequence is periodic with period at most Template:Math.<ref>Template:Citation</ref> The lengths of the periods for various Template:Mvar form the so-called Pisano periods.<ref>Template:Cite OEIS</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 integer or of an element in a finite field. However, for any particular Template:Mvar, the Pisano period may be found as an instance of cycle detection.

GeneralizationsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

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:

ApplicationsEdit

MathematicsEdit

File:Pascal triangle fibonacci.svg
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 coefficients in the "shallow" diagonals of Pascal's triangle:Template:Sfn <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:

Template:Math Template:Math
Template:Math Template:Math Template:Math Template:Math
Template:Math Template:Math Template:Math

which is <math>\textstyle \binom{5}{0}+\binom{4}{1}+\binom{3}{2}</math>, where we are choosing the positions of Template:Mvar twos from Template:Math terms.

File:Fibonacci climbing stairs.svg
Use of the Fibonacci sequence to count Template:Nowrap compositions

These numbers also give the solution to certain enumerative problems,<ref>Template:Citation</ref> the most common of which is that of counting the number of ways of writing a given number Template:Mvar as an ordered sum of 1s and 2s (called compositions); there are Template:Math ways to do this (equivalently, it's also the number of domino tilings of the <math>2\times n</math> rectangle). For example, there are Template:Math ways one can climb a staircase of 5 steps, taking one or two steps at a time:

Template:Math Template:Math Template:Math Template:Math Template:Math Template:Math
Template:Math Template:Math Template:Math

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.

Computer scienceEdit

File:Fibonacci Tree 6.svg
Fibonacci tree of height 6. Balance factors green; heights red.
The keys in the left spine are Fibonacci numbers.

NatureEdit

Template:Further Template:See also

File:FibonacciChamomile.PNG
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>Template:Citation</ref> such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,<ref>Template:Citation</ref> the flowering of artichoke, the arrangement of a pine cone,<ref>Template:Citation</ref> and the family tree of honeybees.<ref>Template:Citation</ref><ref>Template:Citation</ref> Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.Template:Sfn Field daisies most often have petals in counts of Fibonacci numbers.Template:Sfn In 1830, Karl Friedrich Schimper and Alexander Braun discovered that the parastichies (spiral phyllotaxis) of plants were frequently expressed as fractions involving Fibonacci numbers.<ref>Template:Citation</ref>

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.<ref>Template:Citation</ref>

File:SunflowerModel.svg
Illustration of Vogel's model for Template:Math

A model for the pattern of florets in the head of a sunflower was proposed by Template:Ill in 1979.<ref>Template:Citation</ref> This has the form

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

where Template:Mvar is the index number of the floret and Template:Mvar 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 Template:Math, the nearest neighbors of floret number Template:Mvar are those at Template:Math for some index Template:Mvar, which depends on Template:Mvar, 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,Template:Sfn typically counted by the outermost range of radii.<ref>Template:Citation</ref>

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, Template:Math, is the number of female ancestors, which is Template:Math, plus the number of male ancestors, which is Template:Math.<ref>Template:Citation</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
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">Template:Citation</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 Template:Nowrap. 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 Template:Nowrap. 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 Template:Nowrap. Five great-great-grandparents contributed to the male descendant's X chromosome Template:Nowrap, 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
The Fibonacci sequence can also be found in man-made construction, as seen when looking at the staircase inside the Berlin Victory Column.

OtherEdit

  • 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 Template:Mvar reflections, for Template:Math, is the Template:Mvar-th Fibonacci number. (However, when Template:Math, there are three reflection paths, not two, one for each of the three surfaces.)Template:Sfn
  • 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 Template:Mvar being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.<ref>Template:Citation</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>Template:Citation</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">Template:Citation</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.Template:Sfn
  • 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.Template:Sfn See also Template:Slink.

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