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In ], the '''Fibonacci numbers''' are a ] of numbers named after ], known as Fibonacci. Fibonacci's 1202 book '']'' introduced the sequence to Western European mathematics, although the sequence had been previously described in ].<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 ]:

:<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''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;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 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|] 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 ]]]
]
}}
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 ]), in the work of the ] ] (''Chandah-shāstra'', the Art of Prosody, ] or ]). ] was important in ancient Indian ritual because of an emphasis on the purity of utterance. The ] ] (6th century AD) showed how the Fibonacci sequence arose in the analysis of ] with long and short syllables. Subsequently, the ] philosopher ] (c.]) composed a well-known text on these. A commentary on Virahanka's work by ] 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 ]: 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. ] reviews this work in '']'' <!-- see (Vol.&nbsp;1, &sect;1.2.8: Fibonacci Numbers)--> as equivalent formulations of the ] for items of lengths 1 and 2.

In the West, the sequence was first studied by Leonardo of Pisa, known as ], in his ] (])<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'')&nbsp;=&nbsp;''F''(''n''−1)&nbsp;+&nbsp;''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 =] School of Electronics and Physical Sciences}}</ref>

==Relation to the ]==
===Closed form expression===
Like every sequence defined by linear ], the Fibonacci numbers have a ]. It has become known as ]'s formula, even though it was already known by ]:
:<math>F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}={{\varphi^n-(-\varphi)^{-n}} \over {\sqrt 5}}\, ,</math> where <math>\varphi</math> is the ] (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 ]====
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 ] (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 ] of these two series will also satisfy the recursion. These linear combinations form a two-dimensional ]; 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 ], or in terms of the ]:
:<math>F(n)=\bigg\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\bigg\rfloor.</math>

===Limit of consecutive quotients===

] 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 ]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 ] using the Fibonacci rule <math>F(n) = F(n-1) + F(n-2) . \, </math>

==Matrix form==

A 2-dimensional system of linear ] 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 ]s of the matrix A are <math>\varphi\,\!</math> and <math>(1-\varphi)\,\!</math>, and the elements of the ]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 ] of &minus;1, and thus it is a 2&times;2 ]. This property can be understood in terms of the ] 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 &minus;1.

The matrix representation gives the following ] 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 ]
:<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 ]<ref name="dijkstra78">E. W. Dijkstra (1978). ''In honour of Fibonacci.'' </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 ] <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 ].<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 ]s of the ] 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 ] 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 ]
:<math>\bigg</math>
contains a positive integer.<ref>M.&nbsp;Möbius, ''Wie erkennt man eine Fibonacci Zahl?'', Math. Semesterber. (1998) 45; 243–246</ref>

==Identities==
#''F''(''n'' + 1) = ''F''(''n'') + ''F''(''n'' &minus; 1)
#''F''(0) + ''F''(1) + ''F''(2) + … + ''F''(''n'') = ''F''(''n'' + 2) &minus; 1
#''F''(1) + 2 ''F''(2) + 3 ''F''(3) + … + ''n F''(''n'') = ''n F''(''n'' + 2) &minus; ''F''(''n'' + 3) + 2
#''F''(0)² + ''F''(1)² + ''F''(2)² + … + ''F''(''n'')² = ''F''(''n'') ''F''(''n'' + 1)

These identities can be proven using many different methods.
But, among all, we wish to present an elegant proof for each of them using ] here.
In particular, ''F''(''n'') can be interpreted as the number of ways summing 1's and 2's to ''n'' &minus; 1, with the convention that ''F''(0) = 0, meaning no sum will add up to &minus;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.

=== Proof of the first identity ===
], we may assume ''n'' ≥ 1.
Then ''F''(''n'' + 1) counts the number of ways summing 1's and 2's to ''n''.

When the first summand is 1, there are ''F''(''n'') ways to complete the counting for ''n'' &minus; 1; and when the first summand is 2, there are ''F''(''n'' &minus; 1) ways to complete the counting for ''n'' &minus; 2.
Thus, in total, there are ''F''(''n'') + ''F''(''n'' &minus; 1) ways to complete the counting for ''n''.

=== Proof of the second identity ===
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'' &minus; 1.
If the second summand is 2 but the first is 1, then there are ''F''(''n'' &minus; 1) ways to complete the counting for ''n'' &minus; 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'' &minus; 1) + … + ''F''(0) gives the desired counting.

=== Proof of the third identity ===
This identity can be established in two stages.
First, we count the number of ways summing 1s and 2s to &minus;1, 0, …, or ''n'' + 1 such that at least one of the summands is 2.

By our second identity, there are ''F''(''n'' + 2) &minus; 1 ways summing to ''n'' + 1; ''F''(''n'' + 1) &minus; 1 ways summing to ''n''; …; and, eventually, ''F''(2) &minus; 1 way summing to 1.
As ''F''(1) &minus; 1 = ''F''(0) = 0, we can add up all ''n'' + 1 sums and apply the second identity again to obtain
: &nbsp;&nbsp;&nbsp; + + … +
: = + + … + + + ''F''(0)
: = ''F''(''n'' + 2) + &minus; (''n'' + 2)
: = ''F''(''n'' + 2) + &minus; (''n'' + 2)
: = ''F''(''n'' + 2) + ''F''(''n'' + 3) &minus; (''n'' + 3).

On the other hand, we observe from the second identity that there are
* ''F''(0) + ''F''(1) + … + ''F''(''n'' &minus; 1) + ''F''(''n'') ways summing to ''n'' + 1;
* ''F''(0) + ''F''(1) + … + ''F''(''n'' &minus; 1) ways summing to ''n'';
……
* ''F''(0) way summing to &minus;1.
Adding up all ''n'' + 1 sums, we see that there are
* (''n'' + 1) ''F''(0) + ''n'' ''F''(1) + … + ''F''(''n'') ways summing to &minus;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) &minus; (''n'' + 3)

Finally, we complete the proof by subtracting the above identity from ''n'' + 1 times the second identity.

===Identity for doubling ''n''===
There is a very simple formula for doubling ''n'' :<math>F_{2n} = F_{n+1}^2 - F_{n-1}^2 = F_n(F_{n+1}+F_{n-1}) </math>.
<ref></ref>

Another identity useful for calculating ''F<sub>n</sub>'' for large values of ''n'' is
:<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''. ]<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 ], 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 ]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 ], and are useful in setting up the ] to ] a Fibonacci number. Such relations exist in a very general sense for numbers defined by recurrence relations, see the section on multiplication formulae under ]s for details.

==Power series==
The ] of the Fibonacci sequence is the ]
:<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 =]
| 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 ]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 ],
:<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 ]
:<math>\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots</math>

has been proved ] by ].

==Primes and divisibility==
{{main|Fibonacci prime}}
A '''Fibonacci prime''' is a Fibonacci number that is ] {{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 ] runs of ]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 (]).<ref>Ron Knott, .</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 ]: that is,
:](''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>], ''My Numbers, My Friends'', Springer-Verlag 2000</ref><ref>Su, Francis E., et al. , ''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 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 ] <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>] (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<ref>Lemmermeyer, ex. 2.38, pp. 73-74</ref>
<math>
\;\;\;5F^2_{\left(p \pm 1 \right) / 2}
\equiv
\left \{
\begin{array}{cl}
\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{array}
\right.
</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

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

</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 ]. 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&nbsp;+&nbsp;4&nbsp;+&nbsp;3), and 5 (8&nbsp;&minus;&nbsp;3). Skipping 21, the next triangle has sides of length 34, 30 (13&nbsp;+&nbsp;12&nbsp;+&nbsp;5), and 16 (21&nbsp;&minus;&nbsp;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 b = 2 \times 13 \times 21 = 546</math>
:<math>\displaystyle c = 13^2 + 21^2 = 610 \,</math> :<math>\displaystyle c = 13^2 + 21^2 = 610 \,</math>

Revision as of 00:27, 28 May 2008

yamum

b = 2 × 13 × 21 = 546 {\displaystyle \displaystyle b=2\times 13\times 21=546}
c = 13 2 + 21 2 = 610 {\displaystyle \displaystyle c=13^{2}+21^{2}=610\,}
272 2 + 546 2 = 610 2 . {\displaystyle \displaystyle 272^{2}+546^{2}=610^{2}\,.}

Magnitude of Fibonacci numbers

Since F n {\displaystyle F_{n}} is asymptotic to φ n / 5 {\displaystyle \varphi ^{n}/{\sqrt {5}}} , the number of digits in the base b representation of F n {\displaystyle F_{n}\,} is asymptotic to n log b ⁡ φ {\displaystyle n\,\log _{b}\varphi } .

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 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 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").

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.

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.

In music, Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or 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 factor 1.609344 for miles 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 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.

Fibonacci numbers in nature

Sunflower head displaying florets in spirals of 34 and 55 around the outside

Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves.

Przemyslaw Prusinkiewicz advanced the idea that real instances can be in part understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.

A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979. This has the form

θ = 2 π ϕ 2 n {\displaystyle \theta ={\frac {2\pi }{\phi ^{2}}}n} , r = c n {\displaystyle r=c{\sqrt {n}}}

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.

Popular culture

Main article: Fibonacci numbers in popular culture

Since the Fibonacci sequence is easy for amateur mathematicians to understand, there are many examples of the Fibonacci numbers being used in popular culture.

Generalizations

Main article: Generalizations of Fibonacci numbers

The Fibonacci sequence has been generalized in many ways. These include:

  • Extending to negative index n, satisfying Fn = Fn−1 + Fn−2 and, equivalently, F-n = (−1)Fn
  • Generalising the index from positive integers to real numbers using a modification of Binet's formula.
  • Starting with other integers. Lucas numbers have L1 = 1, L2 = 3, and Ln = Ln−1 + Ln−2. Primefree sequences use the Fibonacci recursion with other starting points in order to generate sequences in which all numbers are composite.
  • Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have Pn = 2Pn – 1 + Pn – 2.
  • Not adding the immediately preceding numbers. The Padovan sequence and Perrin numbers 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 with period at most n 2 {\displaystyle n^{2}} . The lengths of the periods for various n form the so-called Pisano periods (sequence A001175 in the OEIS). Determining the Pisano periods in general is an open problem, 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.

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.

See also

References

  1. M. Avriel and D.J. Wilde (1966). "Optimality of the Symmetric Fibonacci Search Technique". Fibonacci Quarterly (3): 265–269.
  2. An Application of the Fibonacci Number Representation
  3. A Practical Use of the Sequence
  4. Zeckendorf representation
  5. S. Douady and Y. Couder (1996). "Phyllotaxis as a Dynamical Self Organizing Process" (PDF). Journal of Theoretical Biology (178): 255–274. doi:10.1006/jtbi.1996.0026.
  6. Jones, Judy (2006). "Science". An Incomplete Education. Ballantine Books. p. 544. ISBN 978-0-7394-7582-9. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  7. A. Brousseau (1969). "Fibonacci Statistics in Conifers". Fibonacci Quarterly (7): 525–532.
  8. Prusinkiewicz, Przemyslaw (1989). Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics). Springer-Verlag. ISBN 0-387-97092-4. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  9. Vogel, H (1979), "A better way to construct the sunflower head", Mathematical Biosciences (44): 179–189
  10. Prusinkiewicz, Przemyslaw (1990). [[The Algorithmic Beauty of Plants]]. Springer-Verlag. pp. 101–107. ISBN 978-0387972978. {{cite book}}: URL–wikilink conflict (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  11. Pravin Chandra and Eric W. Weisstein. "Fibonacci Number". MathWorld.
  12. The Fibonacci Numbers and the Ancestry of Bees
  13. Template:Cite article

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