Fibonacci number
10918
226015045
2008-07-16T13:21:36Z
Silly rabbit
555135
rv: Put it in the External links section, if anywhere. [[WP:UNDO|Undid]] revision 226008684 by [[Special:Contributions/89.146.77.129|89.146.77.129]] ([[User talk:89.146.77.129|talk]])
[[Image:FibonacciBlocks.svg|thumb|180px|right|A tiling with squares whose sides are successive Fibonacci numbers in length]]
[[Image:Yupana 1.GIF|thumb|180px|right|A '''yupana''' ([[Quechua]] for "counting tool") is a calculator which was used by the [[Incas]]. Researchers assume that calculations were based on Fibonacci numbers to minimize the amount of necessary grains per field.<ref>http://www.quipus.it/english/Andean%20Calculators.pdf</ref>]]
In [[mathematics]], the '''Fibonacci numbers''' are a [[sequence]] of numbers named after [[Leonardo of Pisa]], known as Fibonacci. Fibonacci's 1202 book ''[[Liber Abaci]]'' introduced the sequence to Western European mathematics, although the sequence had been previously described in [[Indian mathematics]].<ref>Parmanand Singh. "Acharya Hemachandra and the (so called) Fibonacci Numbers". Math. Ed. Siwan, 20(1):28-30, 1986. ISSN 0047-6269]</ref><ref>Parmanand Singh,"The So-called Fibonacci numbers in ancient and medieval India." Historia Mathematica 12(3), 229–44, 1985.</ref>
The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself. In mathematical terms, it is defined by the following [[recurrence relation]]:
:<math>
F_n =
\begin{cases}
0 & \mbox{if } n = 0; \\
1 & \mbox{if } n = 1; \\
F_{n-1}+F_{n-2} & \mbox{if } n > 1. \\
\end{cases}
</math>
That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers {{OEIS|id=A000045}}, also denoted as ''F<sub>n</sub>'', for ''n'' = 0, 1, 2, … ,20 are:<ref> By modern convention, the sequence begins with ''F''<sub>0</sub>=0. The ''Liber Abaci'' began the sequence with ''F''<sub>1</sub> = 1, omitting the initial 0, and the sequence is still written this way by some.</ref><ref>The website [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibtable.html] has the first 300 F<sub>''n''</sub> factored into primes and links to more extensive tables.</ref>
:{| class="wikitable"
|-
| ''F''<sub>0</sub>
| ''F''<sub>1</sub>
| ''F''<sub>2</sub>
| ''F''<sub>3</sub>
| ''F''<sub>4</sub>
| ''F''<sub>5</sub>
| ''F''<sub>6</sub>
| ''F''<sub>7</sub>
| ''F''<sub>8</sub>
| ''F''<sub>9</sub>
| ''F''<sub>10</sub>
| ''F''<sub>11</sub>
| ''F''<sub>12</sub>
| ''F''<sub>13</sub>
| ''F''<sub>14</sub>
| ''F''<sub>15</sub>
| ''F''<sub>16</sub>
| ''F''<sub>17</sub>
| ''F''<sub>18</sub>
| ''F''<sub>19</sub>
| ''F''<sub>20</sub>
|-
| 0
| 1
| 1
| 2
| 3
| 5
| 8
| 13
| 21
| 34
| 55
| 89
| 144
| 233
| 377
| 610
| 987
| 1597
| 2584
| 4181
| 6765
|}
{{ImageStackRight|200|[[Image:Fibonacci spiral 34.svg|right|thumb|180px|A [[Fibonacci number|Fibonacci spiral]] created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34; see [[Golden spiral]]]]
[[Image:Fibonacci Sequence Plot.PNG|right|thumb|180px|A plot of the Fibonacci sequence from 0 to 1597]]
}}
Every 3rd number of the sequence is even and more generally, every ''k''th number of the sequence is a multiple of ''F<sub>k</sub>''.
The sequence extended to negative index ''n'' satisfies ''F<sub>n</sub>'' = ''F''<sub>''n''−1</sub> + ''F''<sub>''n''−2</sub> for ''all'' integers ''n'', and ''F<sub>−n</sub>'' = (−1)<sup>''n''+1</sup>''F''<sub>''n''</sub>:
.., −8, 5, −3, 2, −1, 1, followed by the sequence above.
==Origins==
The Fibonacci numbers first appeared, under the name ''mātrāmeru'' (mountain of [[cadence]]), in the work of the [[Sanskrit grammarians|Sanskrit grammarian]] [[Pingala]] (''Chandah-shāstra'', the Art of Prosody, [[450 BC|450]] or [[200 BC]]). [[Prosody (linguistics)|Prosody]] was important in ancient Indian ritual because of an emphasis on the purity of utterance. The [[Indian mathematicians|Indian mathematician]] [[Virahanka]] (6th century AD) showed how the Fibonacci sequence arose in the analysis of [[Vedic meter|metres]] with long and short syllables. Subsequently, the [[Jain]] philosopher [[Hemachandra]] (c.[[1150]]) composed a well-known text on these. A commentary on Virahanka's work by [[Gopala (mathematician)|Gopāla]] in the 12th century also revisits the problem in some detail.
Sanskrit vowel sounds can be long (L) or short (S), and Virahanka's analysis, which came to be known as ''mātrā-vṛtta'', wishes to compute how many metres (''mātrā''s) of a given overall length can be composed of these syllables. If the long syllable is twice as long as the short, the solutions are:
: 1 [[mora (linguistics)|mora]]: S (1 pattern)
: 2 morae: SS; L (2)
: 3 morae: SSS, SL; LS (3)
: 4 morae: SSSS, SSL, SLS; LSS, LL (5)
: 5 morae: SSSSS, SSSL, SSLS, SLSS, SLL; LSSS, LSL, LLS (8)
: 6 morae: SSSSSS, SSSSL, SSSLS, SSLSS, SLSSS, LSSSS, SSLL, SLSL, SLLS, LSSL, LSLS, LLSS, LLL (13)
: 7 morae: SSSSSSS, SSSSSL, SSSSLS, SSSLSS, SSLSSS, SLSSSS, LSSSSS, SSSLL, SSLSL, SLSSL, LSSSL, SSLLS, SLSLS, LSSLS, SLLSS, LSLSS, LLSSS, SLLL, LSLL, LLSL, LLLS (21)
A pattern of length ''n'' can be formed by adding S to a pattern of length ''n'' − 1, or L to a pattern of length ''n'' − 2; and the prosodicists showed that the number of patterns of length ''n'' is the sum of the two previous numbers in the sequence. [[Donald Knuth]] reviews this work in ''[[The Art of Computer Programming]]'' <!-- see (Vol. 1, §1.2.8: Fibonacci Numbers)--> as equivalent formulations of the [[bin packing problem]] for items of lengths 1 and 2.
In the West, the sequence was first studied by Leonardo of Pisa, known as [[Fibonacci]], in his [[Liber Abaci]] ([[1202]])<ref>{{cite book | title = Fibonacci's Liber Abaci | author = Sigler, Laurence E. (trans.) | publisher = Springer-Verlag | year = 2002 | id = ISBN 0-387-95419-8}} Chapter II.12, pp. 404–405.</ref>. He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that:
* In the "zeroth" month, there is one pair of rabbits (additional pairs of rabbits = 0)
* In the first month, the first pair begets another pair (additional pairs of rabbits = 1)
* In the second month, both pairs of rabbits have another pair, and the first pair dies (additional pairs of rabbits = 1)
* In the third month, the second pair and the new two pairs have a total of three new pairs, and the older second pair dies. (additional pairs of rabbits = 2)
The laws of this are that each pair of rabbits has 2 pairs in its lifetime, and dies.
Let the population at month ''n'' be ''F''(''n''). At this time, only rabbits who were alive at month ''n'' − 2 are fertile and produce offspring, so ''F''(''n'' − 2) pairs are added to the current population of ''F''(''n'' − 1). Thus the total is ''F''(''n'') = ''F''(''n'' − 1) + ''F''(''n'' − 2).<ref>{{cite web
| last = Knott
| first = Ron
| title = Fibonacci's Rabbits
| url=http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits
| publisher =[[University of Surrey]] School of Electronics and Physical Sciences}}</ref>
==Relation to the [[Golden Ratio]]==
===Closed form expression===
Like every sequence defined by linear [[Recurrence relation|recurrence]], the Fibonacci numbers have a [[closed-form expression|closed-form solution]]. It has become known as [[Jacques Philippe Marie Binet|Binet]]'s formula, even though it was already known by [[Abraham de Moivre]]:
:<math>F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}={{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}\, ,</math> where <math>\varphi</math> is the [[golden ratio]]
:<math>\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,</math>
(note, that <math>1-\varphi=-1/\varphi</math>, as can be seen from the defining equation below).
The Fibonacci recursion
:<math>F(n+2)-F(n+1)-F(n)=0\,</math>
is similar to the defining equation of the golden ratio in the form
:<math>x^2-x-1=0,\,</math>
which is also known as the generating polynomial of the recursion.
====Proof by [[Mathematical induction|induction]]====
Any root of the equation above satisfies <math>\begin{matrix}x^2=x+1,\end{matrix}\,</math> and multiplying by <math>x^{n-1}\,</math> shows:
:<math>x^{n+1} = x^n + x^{n-1}\,</math>
By definition <math>\varphi</math> is a root of the equation, and the other root is <math>1-\varphi=-1/\varphi\, .</math>. Therefore:
:<math>\varphi^{n+1} = \varphi^n + \varphi^{n-1}\, </math>
and
:<math>(1-\varphi)^{n+1} = (1-\varphi)^n + (1-\varphi)^{n-1}\, .</math>
Both <math>\varphi^{n}</math> and <math>(1-\varphi)^{n}=(-1/\varphi)^{n}</math>
are [[geometric series]] (for ''n'' = 1, 2, 3, ...) that satisfy the Fibonacci recursion. The first series grows exponentially; the second exponentially tends to zero, with alternating signs. Because the Fibonacci recursion is linear, any [[linear combination]] of these two series will also satisfy the recursion. These linear combinations form a two-dimensional [[linear vector space]]; the original Fibonacci sequence can be found in this space.
Linear combinations of series <math>\varphi^{n}</math> and <math>(1-\varphi)^{n}</math>, with coefficients ''a'' and ''b'', can be defined by
:<math>F_{a,b}(n) = a\varphi^n+b(1-\varphi)^n</math> for any real <math>a,b\, .</math>
All thus-defined series satisfy the Fibonacci recursion
:<math>\begin{align}
F_{a,b}(n+1) &= a\varphi^{n+1}+b(1-\varphi)^{n+1} \\
&=a(\varphi^{n}+\varphi^{n-1})+b((1-\varphi)^{n}+(1-\varphi)^{n-1}) \\
&=a{\varphi^{n}+b(1-\varphi)^{n}}+a{\varphi^{n-1}+b(1-\varphi)^{n-1}} \\
&=F_{a,b}(n)+F_{a,b}(n-1)\,.
\end{align}</math>
Requiring that <math>F_{a,b}(0)=0</math> and <math>F_{a,b}(1)=1</math> yields <math>a=1/\sqrt 5</math> and <math>b=-1/\sqrt 5</math>, resulting in the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore, an explicit check can be made:
:<math>F_{a,b}(0)=\frac{1}{\sqrt 5}-\frac{1}{\sqrt 5}=0\,\!</math>
and
:<math>F_{a,b}(1)=\frac{\varphi}{\sqrt 5}-\frac{(1-\varphi)}{\sqrt 5}=\frac{-1+2\varphi}{\sqrt 5}=\frac{-1+(1+\sqrt 5)}{\sqrt 5}=1,</math>
establishing the base cases of the induction, proving that
:<math>F(n)={{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}</math> for all <math> n\, .</math>
Therefore, for any two starting values, a combination <math>a,b</math> can be found such that the function <math>F_{a,b}(n)\,</math> is the exact closed formula for the series.
====Computation by rounding====
Since <math>\begin{matrix}|1-\varphi|^n/\sqrt 5 < 1/2\end{matrix}</math> for all <math>n\geq 0</math>, the number <math>F(n)</math> is the closest integer to <math>\varphi^n/\sqrt 5\, .</math> Therefore it can be found by [[Rounding#Rounding_in_an_exact_computation|rounding]], or in terms of the [[floor function]]:
:<math>F(n)=\bigg\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\bigg\rfloor.</math>
===Limit of consecutive quotients===
[[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 the limit approaches the golden ratio <math>\varphi</math>.<ref>{{cite book | last=Kepler | first=Johannes | title=A New Year Gift: On Hexagonal Snow | date=1966 | isbn=0198581203 | publisher=Oxford University Press | pages=92}} Strena seu de Nive Sexangula (1611)</ref>
:<math>\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi,</math>
This convergence does not depend on the starting values chosen, excluding 0, 0.
'''Proof''':
It follows from the explicit formula that for any real <math>a \ne 0, \, b \ne 0 \,</math>
:<math>\begin{align}
\lim_{n\to\infty}\frac{F_{a,b}(n+1)}{F_{a,b}(n)}
&= \lim_{n\to\infty}\frac{a\varphi^{n+1}-b(1-\varphi)^{n+1}}{a\varphi^n-b(1-\varphi)^n} \\
&= \lim_{n\to\infty}\frac{a\varphi-b(1-\varphi)(\frac{1-\varphi}{\varphi})^n}{a-b(\frac{1-\varphi}{\varphi})^n} \\
&= \varphi
\end{align}</math>
because <math>\bigl|{\tfrac{1-\varphi}{\varphi}}\bigr| < 1</math> and thus <math>\lim_{n\to\infty}\left(\tfrac{1-\varphi}{\varphi}\right)^n=0 .</math>
===Decomposition of powers of the golden ratio===
Since the golden ratio satisfies the equation
:<math>\varphi^2=\varphi+1,\,</math>
this expression can be used to decompose higher powers <math>\varphi^n</math> as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of <math>\varphi</math> and 1. The resulting [[recurrence relation]]ships yield Fibonacci numbers as the linear coefficients, thus closing the loop:
:<math>\varphi^n=F(n)\varphi+F(n-1).</math>
This expression is also true for <math>n \, <\, 1 \, </math> if the Fibonacci sequence <math>F(n) \,</math> is [[Generalizations_of_Fibonacci_numbers#Extension_to_negative_integers|extended to negative integers]] using the Fibonacci rule <math>F(n) = F(n-1) + F(n-2) . \, </math>
==Matrix form==
A 2-dimensional system of linear [[difference equations]] that describes the Fibonacci sequence is
:<math>{F_{k+2} \choose F_{k+1}} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k}}</math>
or
:<math>\vec F_{k+1} = A \vec F_{k}.\,</math>
The [[eigenvalue]]s of the matrix A are <math>\varphi\,\!</math> and <math>(1-\varphi)\,\!</math>, and the elements of the [[eigenvector]]s of A, <math>{\varphi \choose 1}</math> and <math>{1 \choose -\varphi}</math>, are in the ratios <math>\varphi\,\!</math> and <math>(1-\varphi\,\!).</math>
This matrix 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:
:<math>\varphi
=1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\;\;\ddots\,}}} \;. </math>
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for <math>\varphi\,\!</math>, and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The matrix representation gives the following [[closed expression]] for the Fibonacci numbers:
:<math>\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n =
\begin{pmatrix} F_{n+1} & F_n \\
F_n & F_{n-1} \end{pmatrix}.
</math>
Taking the determinant of both sides of this equation yields [[Cassini's identity]]
:<math>(-1)^n = F_{n+1}F_{n-1} - F_n^2.\,</math>
Additionally, since <math> A^n A^m=A^{m+n}</math> for any square matrix <math>A</math>, the following identities can be derived:
:<math>{F_n}^2 + {F_{n-1}}^2 = F_{2n-1},\,</math>
:<math>F_{n+1}F_{m} + F_n F_{m-1} = F_{m+n}.\, </math>
For the first one of these, there is a related identity:
:<math>(2F_{n-1}+F_n)F_n = (F_{n-1}+F_{n+1})F_n = F_{2n}.\,</math>
For another way to derive the <math>F_{2n+k}</math> formulas see the "EWD note" by [[Dijkstra]]<ref name="dijkstra78">E. W. Dijkstra (1978). ''In honour of Fibonacci.'' [http://www.cs.utexas.edu/users/EWD/ewd06xx/EWD654.PDF Report EWD654]</ref>.
==Recognizing Fibonacci numbers==
The question may arise whether a positive integer <math>z</math> is a Fibonacci number. Since <math>F(n)</math> is the closest integer to <math>\varphi^n/\sqrt{5}</math>, the most straightforward, brute-force test is the identity
:<math>F\bigg(\bigg\lfloor\log_\varphi(\sqrt{5}z)+\frac{1}{2}\bigg\rfloor\bigg)=z,</math>
which is true [[if and only if]] <math>z</math> is a Fibonacci number.
Alternatively, a positive integer <math>z</math> is a Fibonacci number if and only if one of <math>5z^2+4</math> or <math>5z^2-4</math> is a [[perfect square]].<ref>{{cite book | last=Posamentier | first=Alfred | coauthors = Lehmann, Ingmar| title=The (Fabulous) FIBONACCI Numbers | date=2007 | isbn=978-1-59102-475-0 | publisher=Prometheus Books | pages=305}}</ref>
A slightly more sophisticated test uses the fact that the [[convergent (continued fraction)|convergent]]s of the [[continued fraction]] representation of <math>\varphi</math> are ratios of successive Fibonacci numbers, that is the inequality
:<math>\bigg|\varphi-\frac{p}{q}\bigg|<\frac{1}{q^2}</math>
(with [[coprime]] positive integers <math>p</math>, <math>q</math>) is true if and only if <math>p</math> and <math>q</math> are successive Fibonacci numbers. From this one derives the criterion that <math>z</math> is a Fibonacci number if and only if the [[closed interval]]
:<math>\bigg[\varphi z-\frac{1}{z},\varphi z+\frac{1}{z}\bigg]</math>
contains a positive integer.<ref>M. Möbius, ''Wie erkennt man eine Fibonacci Zahl?'', Math. Semesterber. (1998) 45; 243–246</ref>
==Identities==
Most identities involving Fibonacci numbers draw from [[combinatorial proof|combinatorial arguments]].
''F''(''n'') can be interpreted as the number of ways summing 1's and 2's to ''n'' − 1, with the convention that ''F''(0) = 0, meaning no sum will add up to −1, and that ''F''(1) = 1, meaning the empty sum will "add up" to 0.
Here the order of the summands matters.
For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice.
=== First Identity ===
:<big><math>F_{n+1} = F_{n} + F_{n-1}</math></big>
:''The nth Fibonacci number is the sum of the previous two Fibonacci numbers.''
==== Proof ====
We must establish that the sequence of numbers defined by the combinatorial interpretation above satisfy the same recurrence relation as the Fibonacci numbers (and so are indeed identical to the Fibonacci numbers).
The set of ''F''(''n''+1) ways of making ordered sums of 1's and 2's that sum to ''n'' may be divided into two non-overlapping sets. The first set contains those sums whose first summand is 1; the remainder sums to ''n''−1, so there are ''F''(''n'') sums in the first set. The second set contains those sums whose first summand is 2; the remainder sums to ''n''−2, so there are ''F''(''n''−1) sums in the second set. The first summand can only be 1 or 2, so these two sets exhaust the original set. Thus ''F''(''n''+1) = ''F''(''n'') + ''F''(''n''−1).
=== Second Identity ===
:<math>\sum_{i=0}^n F_i = F_{n+2} - 1</math>
:''The sum of the first n Fibonacci numbers is the (n+2)nd Fibonacci number minus 1.''
==== Proof ====
We count the number of ways summing 1's and 2's to ''n'' + 1 such that at least one of the summands is 2.
As before, there are ''F''(''n'' + 2) ways summing 1's and 2's to ''n'' + 1 when ''n'' ≥ 0.
Since there is only one sum of ''n'' + 1 that does not use any 2, namely 1 + … + 1 (''n'' + 1 terms), we subtract 1 from ''F''(''n'' + 2).
Equivalently, we can consider the first occurrence of 2 as a summand.
If, in a sum, the first summand is 2, then there are ''F''(''n'') ways to the complete the counting for ''n'' − 1.
If the second summand is 2 but the first is 1, then there are ''F''(''n'' − 1) ways to complete the counting for ''n'' − 2.
Proceed in this fashion.
Eventually we consider the (''n'' + 1)th summand.
If it is 2 but all of the previous ''n'' summands are 1's, then there are ''F''(0) ways to complete the counting for 0.
If a sum contains 2 as a summand, the first occurrence of such summand must take place in between the first and (''n'' + 1)th position.
Thus ''F''(''n'') + ''F''(''n'' − 1) + … + ''F''(0) gives the desired counting.
=== Third Identity ===
:<math>\sum_{i=0}^n iF_i = nF_{n+2} - F_{n+3} + 2</math>
==== Proof ====
This identity can be established in two stages.
First, we count the number of ways summing 1s and 2s to −1, 0, …, or ''n'' + 1 such that at least one of the summands is 2.
By our second identity, there are ''F''(''n'' + 2) − 1 ways summing to ''n'' + 1; ''F''(''n'' + 1) − 1 ways summing to ''n''; …; and, eventually, ''F''(2) − 1 way summing to 1.
As ''F''(1) − 1 = ''F''(0) = 0, we can add up all ''n'' + 1 sums and apply the second identity again to obtain
: [''F''(''n'' + 2) − 1] + [''F''(''n'' + 1) − 1] + … + [''F''(2) − 1]
: = [''F''(''n'' + 2) − 1] + [''F''(''n'' + 1) − 1] + … + [''F''(2) − 1] + [''F''(1) − 1] + ''F''(0)
: = ''F''(''n'' + 2) + [''F''(''n'' + 1) + … + ''F''(1) + ''F''(0)] − (''n'' + 2)
: = ''F''(''n'' + 2) + [''F''(''n'' + 3) − 1] − (''n'' + 2)
: = ''F''(''n'' + 2) + ''F''(''n'' + 3) − (''n'' + 3).
On the other hand, we observe from the second identity that there are
* ''F''(0) + ''F''(1) + … + ''F''(''n'' − 1) + ''F''(''n'') ways summing to ''n'' + 1;
* ''F''(0) + ''F''(1) + … + ''F''(''n'' − 1) ways summing to ''n'';
……
* ''F''(0) way summing to −1.
Adding up all ''n'' + 1 sums, we see that there are
* (''n'' + 1) ''F''(0) + ''n'' ''F''(1) + … + ''F''(''n'') ways summing to −1, 0, …, or ''n'' + 1.
Since the two methods of counting refer to the same number, we have
: (''n'' + 1) ''F''(0) + ''n'' ''F''(1) + … + ''F''(''n'') = ''F''(''n'' + 2) + ''F''(''n'' + 3) − (''n'' + 3)
Finally, we complete the proof by subtracting the above identity from ''n'' + 1 times the second identity.
=== Fourth Identity ===
:<math>\sum_{i=0}^n {F_i}^2 = F_{n} F_{n+1}</math>
:''The sum of the first n Fibonacci numbers squared is the product of the nth and (n+1)th Fibonacci numbers.''
=== Identity for doubling ''n'' ===
:<math>F_{2n} = F_{n+1}^2 - F_{n-1}^2 = F_n(F_{n+1}+F_{n-1}) </math>
<ref>[http://mathworld.wolfram.com/FibonacciNumber.html Fibonacci Number - from Wolfram MathWorld<!-- Bot generated title -->]</ref>
=== Another Identity ===
Another identity useful for calculating ''F<sub>n</sub>'' for large values of ''n'' is
:<math>F_{kn+c} = \sum_{i=0}^k {k\choose i} F_{c-i} F_n^i F_{n+1}^{k-i}</math>
<ref>[http://mathworld.wolfram.com/FibonacciNumber.html Fibonacci Number - from Wolfram MathWorld<!-- Bot generated title -->]</ref>
From which other identities for specific values of k, n, and c can be derived below, including
:<math>F_{2n+k} = F_k F_{n+1}^2 + 2 F_{k-1} F_{n+1} F_n + F_{k-2} F_n^2 </math>
for all integers ''n'' and ''k''. [[Dijkstra]]<ref name="dijkstra78"/> points out that doubling identities of this type can be used to calculate ''F<sub>n</sub>'' using O(log ''n'') arithmetic operations. Notice that, with the definition of Fibonacci numbers with negative ''n'' given in the introduction, this formula reduces to the ''double n'' formula when ''k = 0''.
(From practical standpoint it should be noticed that the calculation involves manipulation of numbers with length (number of digits) <math>{\rm \Theta}(n)\,</math>. Thus the actual performance depends mainly upon efficiency of the implemented [[multiplication algorithm| long multiplication]], and usually is <math>{\rm \Theta}(n \,\log n)</math> or <math>{\rm \Theta}(n ^{\log_2 3})</math>.)
===Other identities===
Other identities include relationships to the [[Lucas number]]s, which have the same recursive properties but start with ''L''<sub>''0''</sub>=2 and ''L''<sub>''1''</sub>=1. These properties include
''F''<sub>''2n''</sub>=''F''<sub>''n''</sub>''L''<sub>''n''</sub>.
There are also scaling identities, which take you from ''F''<sub>n</sub> and ''F''<sub>n+1</sub> to a variety of things of the form ''F''<sub>an+b</sub>; for instance
<math>F_{3n} = 2F_n^3 + 3F_n F_{n+1} F_{n-1} = 5F_{n}^3 + 3 (-1)^n F_{n} </math> by Cassini's identity.
<math>F_{3n+1} = F_{n+1}^3 + 3 F_{n+1}F_n^2 - F_n^3</math>
<math>F_{3n+2} = F_{n+1}^3 + 3 F_{n+1}^2F_n + F_n^3</math>
<math>F_{4n} = 4F_nF_{n+1}(F_{n+1}^2 + 2F_n^2) - 3F_n^2(F_n^2 + 2F_{n+1}^2)</math>
These can be found experimentally using [[lattice reduction]], and are useful in setting up the [[special number field sieve]] to [[Factorization|factorize]] a Fibonacci number. Such relations exist in a very general sense for numbers defined by recurrence relations, see the section on multiplication formulae under [[Perrin number]]s for details.
==Power series==
The [[generating function]] of the Fibonacci sequence is the [[power series]]
:<math>s(x)=\sum_{k=0}^{\infty} F_k x^k.</math>
This series has a simple and interesting closed-form solution for <math>|x| < 1/\varphi</math>
:<math>s(x)=\frac{x}{1-x-x^2}.</math>
This solution can be proven by using the Fibonacci recurrence to expand each coefficient in the infinite sum defining <math>s(x)</math>:
:<math>\begin{align}
s(x) &= \sum_{k=0}^{\infty} F_k x^k \\
&= F_0 + F_1x + \sum_{k=2}^{\infty} \left( F_{k-1} + F_{k-2} \right) x^k \\
&= x + \sum_{k=2}^{\infty} F_{k-1} x^k + \sum_{k=2}^{\infty} F_{k-2} x^k \\
&= x + x\sum_{k=0}^{\infty} F_k x^k + x^2\sum_{k=0}^{\infty} F_k x^k \\
&= x + x s(x) + x^2 s(x)
\end{align}</math>
Solving the equation <math>s(x)=x+xs(x)+x^2s(x)</math> for <math>s(x)</math> results in the closed form solution.
In particular, math puzzle-books note the curious value <math>\frac{s(\frac{1}{10})}{10}=\frac{1}{89}</math>, or more generally
:<math>\sum_{n = 1}^{\infty}{\frac {F(n)}{10^{(k + 1)(n + 1)}}} = \frac {1}{10^{2k + 2} - 10^{k + 1} - 1}</math>
for all integers <math>k >= 0</math>.
Conversely,
:<math>\sum_{n=0}^\infty\,\frac{F_n}{k^{n}}\,=\,\frac{k}{k^{2}-k-1}.</math>
==Reciprocal sums==
<!--
{{cite book
| last =Borwein
| first =Jonathan M.
| authorlink =Jonathan Borwein
| coauthors =[[Peter Borwein|Peter B. Borwein]]
| title =Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
| pages =91–101
| publisher =Wiley
| year =1998
| month =July
| url =http://www.wiley.com/WileyCDA/WileyTitle/productCd-047131515X.html
| id = ISBN 978-0-471-31515-5 }}
It credits some formulae to {{cite journal | author = Landau, E. | title = Sur la Série des Invers de Nombres de Fibonacci | journal = Bull. Soc. Math. France | volume = 27 | year = 1899 | pages = 298–300}}
-->
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of [[theta function]]s. For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as
:<math>\sum_{k=0}^\infty \frac{1}{F_{2k+1}} = \frac{\sqrt{5}}{4}\vartheta_2^2 \left(0, \frac{3-\sqrt 5}{2}\right) ,</math>
and the sum of squared reciprocal Fibonacci numbers as
:<math>\sum_{k=1}^\infty \frac{1}{F_k^2} = \frac{5}{24} \left(\vartheta_2^4\left(0, \frac{3-\sqrt 5}{2}\right) - \vartheta_4^4\left(0, \frac{3-\sqrt 5}{2}\right) + 1 \right).</math>
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
:<math>\sum_{k=0}^\infty \frac{1}{1+F_{2k+1}} = \frac{\sqrt{5}}{2},</math>
and there is a nice ''nested'' sum of squared Fibonacci numbers giving the reciprocal of the [[golden ratio]],
:<math>\sum_{k=1}^\infty \frac{(-1)^{k+1}}{\sum_{j=1}^k {F_{j}}^2} = \frac{\sqrt{5}-1}{2}.</math>
Results such as these make it plausible that a closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but none is yet known. Despite that, the [[reciprocal Fibonacci constant]]
:<math>\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots</math>
has been proved [[irrational number|irrational]] by [[Richard André-Jeannin]].
==Primes and divisibility==
{{main|Fibonacci prime}}
A '''Fibonacci prime''' is a Fibonacci number that is [[prime number|prime]] {{OEIS|id=A005478}}. The first few are:
: 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. They must all have a prime index, except ''F''<sub>4</sub> = 3. There are [[Arbitrarily large|arbitrarily long]] runs of [[composite number]]s and therefore also of composite Fibonacci numbers.
With the exceptions of 1, 8 and 144 (''F''<sub>0</sub> = ''F''<sub>1</sub>, ''F''<sub>6</sub> and ''F''<sub>12</sub>) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number ([[Carmichael's theorem]]).<ref>Ron Knott, [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibtable.html "The Fibonacci numbers"].</ref>
No Fibonacci number greater than ''F''<sub>6</sub> = 8 is one greater or one less than a prime number.<ref>Ross Honsberger ''Mathematical Gems III'' (AMS Dolciani Mathematcal Expositions No. 9), 1985, ISBN 0-88385-318-3, p. 133.</ref>
Any three consecutive Fibonacci numbers, taken two at a time, are [[relatively prime]]: that is,
:[[greatest common divisor|gcd]](''F''<sub>''n''</sub>, ''F''<sub>''n''+1</sub>) = gcd(''F''<sub>''n''</sub>, ''F''<sub>''n''+2</sub>) = 1.
More generally,
:gcd(''F''<sub>''n''</sub>, ''F''<sub>''m''</sub>) = ''F''<sub>gcd(''n'', ''m'').</sub><ref>[[Paulo Ribenboim]], ''My Numbers, My Friends'', Springer-Verlag 2000</ref><ref>Su, Francis E., et al. [http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml "Fibonacci GCD's, please."], ''Mudd Math Fun Facts''.</ref>
===Odd divisors===
If ''n'' is odd all the odd divisors of F<sub>''n''</sub> are ≡ 1 (mod 4).<ref>Lemmermeyer, ex. 2.27 p. 73</ref><ref>The website [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibtable.html] has the first 300 Fibonacci numbers factored into primes.</ref> <br>
This is equivalent to saying that for odd ''n'' all the odd prime factors of F<sub>''n''</sub> are ≡ 1 (mod 4).
<blockquote>For example,
F<sub>1</sub> = 1, F<sub>3</sub> = 2, F<sub>5</sub> = 5, F<sub>7</sub> = 13, F<sub>9</sub> = 34 = 2×17, F<sub>11</sub> = 89, F<sub>13</sub> = 233, F<sub>15</sub> = 610 = 2×5×61
</blockquote>
===Fibonacci and Legendre===
There are some interesting formulas connecting the Fibonacci numbers and the [[Legendre symbol]] <math>\;\left(\tfrac{p}{5}\right).</math>
:<math>
\left(\frac{p}{5}\right)
= \left \{
\begin{array}{cl} 0 & \textrm{if}\;p =5
\\ 1 &\textrm{if}\;p \equiv \pm1 \pmod 5
\\ -1 &\textrm{if}\;p \equiv \pm2 \pmod 5
\end{array}
\right.
</math>
If ''p'' is a prime number then<ref>[[Paulo Ribenboim]] (1996), ''The New Book of Prime Number Records'', New York: Springer, ISBN 0-387-94457-5, p. 64</ref><ref>Franz Lemmermeyer (2000), ''Reciprocity Laws'', New York: Springer, ISBN 3-540-66957-4, ex 2.25-2.28, pp. 73-74</ref>
<math>
F_{p} \equiv \left(\frac{p}{5}\right) \pmod p \;\;\mbox{ and }\;\;\;
F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p.
</math>
<blockquote>
For example,
:<math>(\tfrac{2}{5}) = -1, \,\, F_3 = 2, F_2=1,</math>
:<math>(\tfrac{3}{5}) = -1, \,\, F_4 = 3,F_3=2,</math>
:<math>(\tfrac{5}{5}) = \;\;\,0,\,\, F_5 = 5,</math>
:<math>(\tfrac{7}{5}) = -1, \,\,F_8 = 21,\;\;F_7=13,</math>
:<math>(\tfrac{11}{5}) = +1, F_{10} = 55, F_{11}=89.</math>
</blockquote>
Also, if ''p'' ≠ 5 is an odd prime number then: <ref>Lemmermeyer, ex. 2.38, pp. 73-74</ref>
:<math>5F^2_{\left(p \pm 1 \right) / 2}
\equiv
\begin{cases}
\frac{5\left(\frac{p}{5}\right)\pm 5}{2} \pmod p & \textrm{if}\;p \equiv 1 \pmod 4\\
\\
\frac{5\left(\frac{p}{5}\right)\mp 3}{2} \pmod p & \textrm{if}\;p \equiv 3 \pmod 4
\end{cases}
</math>
<blockquote>
Examples of all the cases:
:<math>p=7 \equiv 3 \pmod 4, \;\;(\tfrac{7}{5}) = -1, \frac{5(\frac{7}{5})+3}{2} =-1\mbox{ and }\frac{5(\frac{7}{5})-3}{2}=-4.</math>
::<math>F_3=2 \mbox{ and } F_4=3.</math>
::<math>5F_3^2=20\equiv -1 \pmod {7}\;\;\mbox{ and }\;\;5F_4^2=45\equiv -4 \pmod {7}</math>
:<math>p=11 \equiv 3 \pmod 4, \;\;(\tfrac{11}{5}) = +1, \frac{5(\frac{11}{5})+3}{2} =4\mbox{ and }\frac{5(\frac{11}{5})- 3}{2}=1.</math>
::<math>F_5=5 \mbox{ and } F_6=8.</math>
::<math>5F_5^2=125\equiv 4 \pmod {11} \;\;\mbox{ and }\;\;5F_6^2=320\equiv 1 \pmod {11}</math>
:<math>p=13 \equiv 1 \pmod 4, \;\;(\tfrac{13}{5}) = -1, \frac{5(\frac{13}{5})-5}{2} =-5\mbox{ and }\frac{5(\frac{13}{5})+ 5}{2}=0.</math>
::<math>F_6=8 \mbox{ and } F_7=13.</math>
::<math>5F_6^2=320\equiv -5 \pmod {13} \;\;\mbox{ and }\;\;5F_7^2=845\equiv 0 \pmod {13}</math>
:<math>p=29 \equiv 1 \pmod 4, \;\;(\tfrac{29}{5}) = +1, \frac{5(\frac{29}{5})-5}{2} =0\mbox{ and }\frac{5(\frac{29}{5})+5}{2}=5.</math>
::<math>F_{14}=377 \mbox{ and } F_{15}=610.</math>
::<math>5F_{14}^2=710645\equiv 0 \pmod {29} \;\;\mbox{ and }\;\;5F_{15}^2=1860500\equiv 5 \pmod {29}</math>
</blockquote>
===Divisibility by 11===
<math>\sum_{k=n}^{n+9} F_{k} = 11 F_{n+6}</math>
<blockquote>For example, let ''n'' = 1:
<br>
F<sub>1</sub>+F<sub>2</sub>+...+F<sub>10</sub> = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11×13
<br>
''n'' = 2:
<br>
F<sub>2</sub>+F<sub>3</sub>+...+F<sub>11</sub> = 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 = 231 = 11×21
<br>
''n'' = 3:
<br>
F<sub>3</sub>+F<sub>4</sub>+...+F<sub>12</sub> = 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144= 374 = 11×34
<br><br>
In fact, the identity is true for all integers ''n'', not just positive ones:<br>
''n'' = 0:
<br>
F<sub>0</sub>+F<sub>1</sub>+...+F<sub>9</sub> = 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 = 88 = 11×8
<br>
''n'' = −1:
<br>
F<sub>−1</sub>+F<sub>0</sub>+...+F<sub>8</sub> = 1 + 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 = 55 = 11×5
<br>
''n'' = −2:
<br>
F<sub>−2</sub>+F<sub>−1</sub>+F<sub>0</sub>+...+F<sub>7</sub> = −1 + 1 + 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 = 33 = 11×3
<br>
</blockquote>
==Right triangles==
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]]. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.
The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely. The triangle sides a, b, c can be calculated directly:
:<math>\displaystyle a_n = F_{2n-1}</math>
:<math>\displaystyle b_n = 2 F_n F_{n-1}</math>
:<math>\displaystyle c_n = {F_n}^2 - {F_{n-1}}^2</math>
These formulas satisfy <math>a_n ^2 = b_n ^2 + c_n ^2</math> for all n, but they only represent triangle sides when <math>n > 2</math>.
Any four consecutive Fibonacci numbers ''F''<sub>''n''</sub>, ''F''<sub>''n''+1</sub>, ''F''<sub>''n''+2</sub> and ''F''<sub>''n''+3</sub> can also be used to generate a Pythagorean triple in a different way:
:<math> a = F_n F_{n+3} \, ; \, b = 2 F_{n+1} F_{n+2} \, ; \, c = F_{n+1}^2 + F_{n+2}^2 \, ; \, a^2 + b^2 = c^2 \,.</math>
Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then:
:<math>\displaystyle a = 1 \times 5 = 5</math>
:<math>\displaystyle b = 2 \times 2 \times 3 = 12</math>
:<math>\displaystyle c = 2^2 + 3^2 = 13 \,</math>
:<math>\displaystyle 5^2 + 12^2 = 13^2 \,.</math>
Example 2: let the Fibonacci numbers be 8, 13, 21 and 34. Then:
:<math>\displaystyle a = 8 \times 34 = 272</math>
:<math>\displaystyle b = 2 \times 13 \times 21 = 546</math>
:<math>\displaystyle c = 13^2 + 21^2 = 610 \,</math>
:<math>\displaystyle 272^2 + 546^2 = 610^2 \,.</math>
==Magnitude of Fibonacci numbers==
Since <math>F_n</math> is [[asymptotic]] to <math>\varphi^n/\sqrt5</math>, the number of digits in the base ''b'' representation of <math>F_n\,</math> is asymptotic to <math>n\,\log_b\varphi</math>.
In base 10, for every integer greater than 1 there are 4 or 5 Fibonacci numbers with that number of digits, in most cases 5.
==Applications==
The Fibonacci numbers are important in the run-time analysis of [[Euclidean algorithm|Euclid's algorithm]] to determine the [[greatest common divisor]] of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
[[Yuri Matiyasevich]] was able to show that the Fibonacci numbers can be defined by a [[Diophantine equation]], which led to [[Matiyasevich's theorem|his original solution]] of [[Hilbert's tenth problem]].
The Fibonacci numbers occur in the sums of "shallow" diagonals in [[Pascal's triangle]] and [[Lozanić's triangle]] (''see "[[Binomial coefficient]]"''). (They occur more obviously in [[Hosoya's triangle]]).
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 Fibonacci numbers and principle is also used in the [[financial markets]]. It is used in trading algorithms, applications and strategies. Some typical forms include: the Fibonacci fan, Fibonacci Arc, Fibonacci Retracement and the Fibonacci Time Extension.
Fibonacci numbers are used by some [[pseudorandom number generators]].<!-- Knuth vol. 2 -->
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 φ. A tape-drive implementation of the polyphase merge sort was described in ''[[The Art of Computer Programming]]''.
Fibonacci numbers arise in the analysis of the [[Fibonacci heap]] data structure.
A one-dimensional optimization method, called the [[Fibonacci search technique]], uses Fibonacci numbers.<ref>{{cite journal | author=M. Avriel and D.J. Wilde | title=Optimality of the Symmetric Fibonacci Search Technique | journal=[[Fibonacci Quarterly]] | year=1966 | issue=3 | pages= 265–269}}</ref>
In [[music]], Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of [[content]] or [[form (music)|formal]] elements. It is commonly thought that the first movement of [[Béla Bartók]]'s ''[[Music for Strings, Percussion, and Celesta]]'' was structured using Fibonacci numbers.
Since the [[conversion of units|conversion]] factor 1.609344 for [[mile]]s to kilometers is close to the [[golden ratio]] (denoted φ), 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 [[Fibonacci coding|number]] [[processor register|register]] in [[golden ratio base]] φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.<ref>[http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrep.html#kilos An Application of the Fibonacci Number Representation]</ref><ref>[http://people.bath.ac.uk/pst20/fibonacci.html#Sequence A Practical Use of the Sequence]</ref><ref>[http://eom.springer.de/Z/z120020.htm Zeckendorf representation]</ref>
==Fibonacci numbers in nature==
[[Image:Helianthus whorl.jpg|thumb|[[Sunflower]] head displaying florets in spirals of 34 and 55 around the outside]]
Fibonacci sequences appear in biological settings,<ref>{{cite journal | author=S. Douady and Y. Couder | title=Phyllotaxis as a Dynamical Self Organizing Process | journal=Journal of Theoretical Biology | year=1996 | issue=178 | pages= 255–274 | url=http://www.math.ntnu.no/~jarlet/Douady96.pdf | doi = 10.1006/jtbi.1996.0026 | volume=178}}</ref> in two consecutive Fibonacci numbers, such as branching in trees, arrangement of [[leaves]] on a stem, the fruitlets of a [[pineapple]],<ref>{{cite book|first=Judy|last=Jones|coauthors=William Wilson|title=An Incomplete Education|publisher=Ballantine Books|year=2006|id=ISBN 978-0-7394-7582-9|pages=544|chapter=Science}}</ref> the flowering of [[artichoke]], an uncurling fern and the arrangement of a [[pine cone]].<ref>{{cite journal | author=A. Brousseau | title=Fibonacci Statistics in Conifers | journal=[[Fibonacci Quarterly]] | year=1969 | issue=7 | pages= 525–532}}</ref> In addition, numerous poorly substantiated claims of Fibonacci numbers or [[golden section]]s in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves{{Fact|date=February 2007}}. The Fibonacci numbers are also found in the family tree of honeybees. <ref>http://www.cs4fn.org/maths/bee-davinci.php</ref>
[[Przemyslaw Prusinkiewicz]] advanced the idea that real instances can be in part understood as the expression of certain algebraic constraints on [[free group]]s, specifically as certain [[L-system|Lindenmayer grammar]]s.<ref>{{cite book|first=Przemyslaw|last=Prusinkiewicz|coauthors=James Hanan|title=Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics)|publisher=[[Springer Science+Business Media|Springer-Verlag]]|year=1989|id=ISBN 0-387-97092-4}}</ref>
A model for the pattern of [[floret]]s in the head of a [[sunflower]] was proposed by H. Vogel in 1979.<ref>
{{Citation
| last =Vogel
| first =H
| title =A better way to construct the sunflower head
| journal =Mathematical Biosciences
| issue =44
| pages =179–189
| year =1979
| doi =10.1016/0025-5564(79)90080-4
| volume =44
}}</ref>
This has the form
:<math>\theta = \frac{2\pi}{\phi^2} n</math>, <math>r = c \sqrt{n}</math>
where ''n'' is the index number of the floret and ''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 F(j):F(j+1), the nearest neighbors of floret number ''n'' are those at ''n''±F(j) for some index ''j'' which depends on ''r'', the distance from the center. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.<ref>{{cite book
| last =Prusinkiewicz
| first =Przemyslaw
| authorlink =Przemyslaw Prusinkiewicz
| coauthors =[[Aristid Lindenmayer|Lindenmayer, Aristid]]
| title =[[The Algorithmic Beauty of Plants]]
| publisher =Springer-Verlag
| date= 1990
| location =
| pages =101-107
| url =http://algorithmicbotany.org/papers/#webdocs
| doi =
| id = ISBN 978-0387972978 }}</ref>
== Popular culture ==
{{main|Fibonacci numbers in popular culture}}
<!--NOTE: YOUR FAVOURITE FIBONACCI REFERENCE SHOULD ONLY BE IN MAIN ARTICLE (Fibonacci numbers in popular culture) AND MAY ALREADY BE THERE!-->
==Generalizations==
{{main|Generalizations of Fibonacci numbers}}
The Fibonacci sequence has been generalized in many ways. These 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]]. <ref>{{MathWorld|title=Fibonacci Number|urlname=FibonacciNumber|author=Pravin Chandra and [[Eric W. Weisstein]]}}</ref>
* Starting with other integers. [[Lucas number]]s have ''L''<sub>1</sub> = 1, ''L''<sub>2</sub> = 3, and ''L<sub>n</sub>'' = ''L''<sub>''n''−1</sub> + ''L''<sub>''n''−2</sub>. [[Primefree sequence]]s use the Fibonacci recursion with other starting points in order to generate sequences in which all numbers are [[composite number|composite]].
* Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The [[Pell number]]s have ''P<sub>n</sub>'' = 2''P''<sub>''n'' – 1</sub> + ''P''<sub>''n'' – 2</sub>.
* Not adding the immediately preceding numbers. The [[Padovan sequence]] and [[Perrin number]]s have P(n) = P(n – 2) + P(n – 3).
* Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more.
* Adding other objects than integers, for example functions or strings -- one essential example is [[Fibonacci polynomials]].
==Numbers properties==
===Periodicity mod ''n'': Pisano periods===
It is easily seen that if the members of the Fibonacci sequence are taken mod ''n'', the resulting sequence must be [[periodic sequence|periodic]] with period at most <math>n^2</math>. The lengths of the periods for various ''n'' form the so-called [[Pisano period]]s {{OEIS|id=A001175}}. Determining the Pisano periods in general is an open problem,{{Fact|date=March 2008}} although for any particular ''n'' it can be solved as an instance of [[cycle detection]].
==The bee ancestry code==
Fibonacci numbers also appear in the description of the reproduction of a population of idealized bees, according to the following rules:
*If an egg is laid by an unmated female, it hatches a male.
*If, however, an egg was fertilized by a male, it hatches a female.
Thus, a male bee will always have one parent, and a female bee will have two.
If one traces the ancestry of any male bee (1 bee), he has 1 female parent (1 bee). This female had 2 parents, a male and a female (2 bees). The female had two parents, a male and a female, and the male had one female (3 bees). Those two females each had two parents, and the male had one (5 bees). This sequence of numbers of parents is the Fibonacci sequence.<ref>[http://american-university.com/cas/mathstat/newstudents/shared/puzzles/fibbee.html The Fibonacci Numbers and the Ancestry of Bees]</ref>
This is an idealization that does not describe ''actual'' bee ancestries. In reality, some ancestors of a particular bee will always be sisters or brothers, thus breaking the lineage of distinct parents.
==Miscellaneous==
In 1963, John H. E. Cohn proved that the only squares among the Fibonacci numbers are 0, 1, and 144.<ref>{{cite article | title=Square Fibonacci Numbers Etc |author= J H E Cohn |journal= Fibonacci Quarterly | volume= 2 | year= 1964 | pages=109-113 | url= http://math.la.asu.edu/~checkman/SquareFibonacci.html}}</ref>
==See also==
*[[Logarithmic spiral]]
*[[b:Fibonacci number program|Fibonacci number program]] at [[Wikibooks]]
*[[The Fibonacci Association]]
*[[Fibonacci Quarterly]] — an academic journal devoted to the study of Fibonacci numbers
*[[Negafibonacci]] numbers
*[[Lucas number]]
*[[Fibonacci fractal]]
==References==
{{reflist|2}}
==External links==
{{external links}}
* Peter Marcer, ''[http://golden-ratio-in-dna.blogspot.com/2008/01/1992-order-and-chaos-in-dnathe-denis.html describing the discovery by jean-claude Perez of Fibonacci numbers structuring proportions of TCAG nucleotides within DNA]'', (1992).
* Ron Knott, ''[http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html The Golden Section: Phi]'', (2005).
* Ron Knott, ''[http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrep.html Representations of Integers using Fibonacci numbers]'', (2004).
* wallstreetcosmos.com, ''[http://www.wallstreetcosmos.com/elliot.html Fibonacci numbers and stock market analysis]'', (2008).
* Juanita Lofthouse ''[http://arxiv.org/abs/physics/0411169 Fibonacci numbers and Red Blood Cell Dynamics]'', .
* Bob Johnson, ''[http://www.dur.ac.uk/bob.johnson/fibonacci/ Fibonacci resources]'', (2004)
* Donald E. Simanek, ''[http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm Fibonacci Flim-Flam]'', (undated, 2005 or earlier).
* Rachel Hall, ''[http://www.sju.edu/~rhall/Multi/rhythm2.pdf Hemachandra's application to Sanskrit poetry]'', (undated; 2005 or earlier).
* Alex Vinokur, ''[http://semillon.wpi.edu/~aofa/AofA/msg00012.html Computing Fibonacci numbers on a Turing Machine]'', (2003).
* (no author given), ''[http://www.goldenmeangauge.co.uk/fibonacci.htm Fibonacci Numbers Information]'', (undated, 2005 or earlier).
* [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html Fibonacci Numbers and the Golden Section] – Ron Knott's Surrey University multimedia web site on the Fibonacci numbers, the Golden section and the Golden string.
* The [http://www.mscs.dal.ca/Fibonacci/ Fibonacci Association] incorporated in [[1963]], focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas.
* Dawson Merrill's [http://www.goldenratio.org/info/ Fib-Phi] link page.
* [http://primes.utm.edu/glossary/page.php?sort=FibonacciPrime Fibonacci primes]
* [http://www.mathpages.com/home/kmath078.htm Periods of Fibonacci Sequences Mod m] at MathPages
* [http://www.upl.cs.wisc.edu/~bethenco/fibo/ The One Millionth Fibonacci Number]
* [http://www.bigzaphod.org/fibonacci/ The Ten Millionth Fibonacci Number]
* An [http://www.calcresult.com/maths/Sequences/expanded_fibonacci.html Expanded Fibonacci Series Generator]
* Manolis Lourakis, [http://www.ics.forth.gr/~lourakis/fibsrch/ Fibonaccian search in C]
* [http://www.physorg.com/news97227410.html Scientists find clues to the formation of Fibonacci spirals in nature]
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=630&bodyId=1002 Fibonacci Numbers] at [http://mathdl.maa.org/convergence/1/ Convergence]
* [http://www.tools4noobs.com/online_tools/fibonacci/ Online Fibonacci calculator]
[[Category:Fibonacci numbers|*]]
[[Category:Articles containing proofs]]
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[[ar:متتالية فيبوناتشي]]
[[az:Fibonnaççi ədədləri]]
[[bn:ফিবোনাচ্চি রাশিমালা]]
[[bs:Fibonaccijev broj]]
[[bg:Числа на Фибоначи]]
[[ca:Successió de Fibonacci]]
[[cs:Fibonacciho posloupnost]]
[[da:Fibonacci-tal]]
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[[el:Ακολουθία Φιμπονάτσι]]
[[et:Fibonacci jada]]
[[es:Sucesión de Fibonacci]]
[[eo:Fibonaĉi-nombro]]
[[eu:Fibonacciren zenbakiak]]
[[fa:اعداد فیبوناچی]]
[[fr:Suite de Fibonacci]]
[[ko:피보나치 수]]
[[hi:हेमचन्द्र श्रेणी]]
[[hr:Fibonaccijev broj]]
[[id:Bilangan Fibonacci]]
[[is:Fibonacciruna]]
[[it:Successione di Fibonacci]]
[[he:סדרת פיבונאצ'י]]
[[lv:Fibonači skaitļi]]
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[[hu:Fibonacci-számok]]
[[ml:ഫിബനാച്ചി സംഖ്യ]]
[[nl:Rij van Fibonacci]]
[[ja:フィボナッチ数]]
[[no:Fibonaccifølgen]]
[[pms:Sequensa ëd Fibonacci]]
[[pl:Ciąg Fibonacciego]]
[[pt:Número de Fibonacci]]
[[ro:Numerele Fibonacci]]
[[ru:Числа Фибоначчи]]
[[scn:Succissioni di Fibonacci]]
[[sk:Fibonacciho postupnosť]]
[[sl:Fibonaccijevo število]]
[[sr:Фибоначијев низ]]
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[[ta:ஃபிபனாச்சி எண்கள்]]
[[th:เลขฟีโบนัชชี]]
[[vi:Dãy Fibonacci]]
[[tr:Fibonacci serisi]]
[[uk:Послідовність Фібоначчі]]
[[vls:Reke van Fibonacci]]
[[zh:斐波那契数列]]