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{{pelbagai isu|{{cleanup|reason=memerlukan penterjemahan segera kerana sudah ditinggalkan sejak tahun 2008|date=Ogos 2014}}{{Terjemah|en|fabonacci number|date=Ogos 2014}}}}
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[[Image:FibonacciBlocks.svg|thumb|180px|right|Suatu ubinan dengan segi empat yang tepinya adalah nombor Fibonaci berturut-turut pada panjangnya]]
[[Image:Yupana 1.png|thumb|180px|right|Sebuah '''yupana''' ([[Quechua]] untuk "alat pengiraan") adalah sebuah kalkulator yang digunakan oleh [[Incas]]. Pengaji menganggapkan bahawa pengiraan adalah berasaskan nombor Fibonacci untuk mengurangkan bilangan biji yang diperlukan tiap sawah.<ref>http://www.quipus.it/english/Andean%20Calculators.pdf</ref>]]
[[Image:Fibonacci spiral 34.svg|right|thumb|180px|Sebuah [[Nombor Fibonacci|Lingkaran Fibonacci]] dicipta dengan melukis lengkung menyambungkan sudut berlawan segi empat dalam ubinan Fibonacci; yang ini menggunakan segi empat-segi empat pada saiz 1, 1, 2, 3, 5, 8, 13, 21, dand   34; see [[Golden spiral]]]]
Dalam [[matematik]], '''nombor Fibonacci''' adalah suatu [[langkah]] nombor dinamakan sempena [[Leonardo of Pisa]], digelar sebagai Fibonacci. Buku 1202 ''[[Liber Abaci]]'' Fibonacci memperkenalkan urutannya ke matematik Eropah Barat, walaupun urutannya telah terdahulu dijelaskan pada [[matematik India]].<ref>Parmanand Singh. "Acharya Hemachandra and the (so called) Fibonacci Numbers". Math. Ed. Siwan, 20(1):28-30(contracted; show full)

Satu corak panjang ''n'' boleh dibentuk dengan menambah S kepada corak panjang ''n''&nbsp;−&nbsp;1, atau L kepada corak panjang ''n''&nbsp;−&nbsp;2; dan pakar prosodi menunjukkan bahawa bilangan corak panjang ''n'' adalah jumlah dua nombor sebelumnya dalam urutan.  [[Donald Knuth]] menyemak kerja ini dalam 
  ''[[The Art of Computer Programming|Art of Computer Programming]]'' <!-- see (Vol.&nbsp;1, &sect;1.2.8: Fibonacci Numbers)--> sebagai rumusan bersamaan [[masalah bin packing]] item dengan panjang 1 dan 2.

(contracted; show full)  | 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>

== Kaitannya dengan [[Nisbah Emas]] ==
===Closed form expression===
Like 
everysetiap sequence defined by linear [[Recurrence relation|recurrence]], the nombor Fibonacci numbers have mempunyaia  [[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|nisbah keemasan]]  
:<math>\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,</math>  {{OEIS|id=A001622}}
(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 nisbah keemasan dalam form

:<math>x^2-x-1=0,\,</math>

which is also known as the generating polyinomial penjana 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 dan   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>

BothKedua-dua <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 Jujukan 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 untuk sebaranyg 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 thedalam 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 untuk sebaranyg 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.

====Pengiraan melalui pembundaran====
Memandangkan <math>\begin{matrix}|1-\varphi|^n/\sqrt 5 < 1/2\end{matrix}</math> bagi semua <math>n\geq 0</math>, nombor <math>F(n)</math> adalah integer yang paling hampir dengan <math>\varphi^n/\sqrt 5\, .</math> Oleh itu, ia boleh didapati dengan [[Pembundaran#Pembundaran dalam pengiraan tepat|pembundaran]], atau dari segi [[fungsi lantai]]:
:<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 ofmemerhatikan bahawa nisbah consecutive nombor 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 nisbah keemasan <math>\varphi</math>.<ref>{{cite book | last=Kepler | first=Johannes | title=A New Year Gift: On Hexagonal Snow | year=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 untuk sebaranyg 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 nisbah keemasan===
Since  nisbah keemasan 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|recurrence hubungan]] yield nombor Fibonacci as  linear coefficients, thus closing the loop:
:<math>\varphi^n=F(n)\varphi+F(n-1).</math>
⏎
⏎
This expression is also truebenar for <math>n \, <\, 1 \, </math> if the jujukan 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 formBentuk matriks==

A 2-dimensional system of linear [[difference equations]] that describes the Jujukan 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 matrixks 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 ratiosdalam nisbah-nisbah <math>\varphi\,\!</math> and <math>(1-\varphi\,\!).</math>

This matrix has amatriks ini mempunyai [[determinant]] of &minus;1, and thus it is a 2&times;2 [[unimodular matrix|unimodular matriks]].  This property can be understood in terms of the [[continued fractio [[pecahan berterusan]] representation for the golden ratio nisbah keemasan: 
:<math>\varphi
=1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\;\;\ddots\,}}} \;. </math> 
The⏎
Nombor Fibonacci numbers occur as the ratio of successive  convergents of the continued fraction fornisbah pertembungan pecahan berterusan yang berterusan <math>\varphi\,\!</math>, dand the matrix formedks yang dibentuk daripada from successive convergents of any continued fraction has apecahan berterusan mempunyai determinant of +1 or &minus;1.

The matrix representation gives the following [[closed expression]] for the Fibonacci numbersPerwakilan matriks memberikan [[ungkapan tertutup]] nombor Fibonacci yang berikut:
:<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 identitykedua-dua belah persamaan ini menyerlahkan [[identiti Cassini]]
⏎
⏎
:<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 untuk sebaranyg square matrixks <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 theUntuk 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 ⏎
Untukanother 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 Mengenalpasti nombor Fibonacci==⏎
⏎
question may arise whether a positivesama ada sesuatu integer positif <math>z</math> is a nombor 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 ifbenar [[jika dan hanya sekiranya]] <math>z</math> is amerupakan nombor 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 | year=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 integerinteger positif <math>z</math> ialah nombor Fibonacci sekiranya dan hanya sekiranya salah satu <math>5z^2+4</math> or <math>5z^2-4</math> merupakan [[segi empat yang sempurna]].<ref>{{cite book | last=Posamentier | first=Alfred | coauthors = Lehmann, Ingmar| title=The (Fabulous) FIBONACCI Numbers | year=2007 | isbn=978-1-59102-475-0 | publisher=Prometheus Books | pages=305}}</ref> 

A slightly more sophisticated test uses  fact that  [[convergent (continued fraction)|convergent]]s perwakilan [[pecahan berterusan]] <math>\varphi</math> ialah nisbah-nisbah nombor Fibonacci yang berturutan, that is  inequality
:<math>\bigg|\varphi-\frac{p}{q}\bigg|<\frac{1}{q^2}</math>
(with [[coprime]] integer positif <math>p</math>, <math>q</math>) is benar if and only if <math>p</math> and <math>q</math> are successive nombor Fibonacci. From this one derives  criterion that <math>z</math> is a nombor Fibonacci if and only if  [[closed interval]]
:<math>\bigg[\varphi z-\frac{1}{z},\varphi z+\frac{1}{z}\bigg]</math>

contains a integer positif.<ref>M.&nbsp;Möbius, ''Wie erkennt man eine Fibonacci Zahl?'', Math. Semesterber. (1998) 45; 243–246</ref>

==Pengenalan==

Kebanyakan pengenalan melibatkan nombor Fibonacci menarik dari [[bukti kombinatorik|hujah kombinatorik]].
(contracted; show full)=== Pengenalan bagi ''n'' berganda ===

:<math>F_{2n} = F_{n+1}^2 - F_{n-1}^2 = F_n(F_{n+1}+F_{n-1}) </math>

<ref name="autogenerated1">[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'' isuntuk mengira  ''F<sub>n</sub>'' memperoleh nilai besar ''n'' ialah

:<math>F_{kn+c} = \sum_{i=0}^k {k\choose i} F_{c-i} F_n^i F_{n+1}^{k-i},</math> 
<ref name="autogenerated1" />

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 definition of nombor Fibonacci with negative ''n'' given dalam introduction, this formula reduces to  ''double n'' formula apabila ''k = 0''.

(From practical standpoint it should be noticed that  calculation involves manipulation of numbers with length (number of digits) <math>{\rm \Theta}(n)\,</math>. Thus  actual performance depends mainly upon efficiency of  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 termasuk  hubungan kepada [[nombor Lucas]] yang mempunyai same recursive properties but start with ''L''<sub>''0''</sub>=2 and ''L''<sub>''1''</sub>=1. These properties termasuk ''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  form ''F''<sub>an+b</sub>; sebagai contoh

<math>F_{3n} = 2F_n^3 + 3F_n F_{n+1} F_{n-1} = 5F_{n}^3 + 3 (-1)^n F_{n} </math> oleh identiti Cassini.

<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|lattice pengurangan]], and are useful in setting up  [[special number field sieve]] to [[Factorization|factorize]] a nombor Fibonacci. Such relations exist in a very general sense for numbers defined by recurrence relations, see  section on multiplication formulae under [[Perrin number]]s for details.

==Siri kuasa==
[[Fungsi generasi]] urutan Fibonacci adalah [[siri kuasa]]
:<math>s(x)=\sum_{k=0}^{\infty} F_k x^k.</math>

Siri ini adalah mudah dan jawapan bentuk-tertutup menarik untuk <math>|x| < 1/\varphi</math>
:<math>s(x)=\frac{x}{1-x-x^2}.</math>
(contracted; show full)
:<math>\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots</math> 

telah dibuktikan [[nombor tak nisbah|tidak bernisbah]] oleh [[Richard André-Jeannin]].

==Nombor perdana dan kebolehbahagian==
{{main|Fibonacci perdana}}
'''Fibonacci perdana''' adalah nombor Fibonacci yang [[nombor perdana|perdana ]] {{OEIS|id=A005478}}. 
The first few are:
: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …
Fibonacci perdana dengan beribu-ribu digit telah ditemui, tetapi tidak diketahui sama ada ia terdapat banyak tak terhingga. Semuanya mesti mempunyai indeks utama, kecuali ''F''<sub>4</sub> = 3. Terdapat aturan [[besar sembarangan|yang sewenang-wenangnya panjang]] bagi [[nombor komposit]] dan dengan itu termasuk juga nombor Fibonacci komposit.

(contracted; show full)''n'' = &minus;2:

<br>
F<sub>&minus;2</sub>+F<sub>&minus;1</sub>+F<sub>0</sub>+...+F<sub>7</sub> = &minus;1 + 1 + 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 = 33 = 11×3

</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&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 numbersSegi tiga bersudut tegak==
Starting with 5, setiap second nombor Fibonacci is  length of  hypotenuse of a right segi tiga with integer sides, or in other words, nombor terbesar dalam  [[ganda tiga Pythagoras]].   length of  longer leg of this segi tiga is equal to  sum of  three sides of  preceding segi tiga in this series of segi tigas, and  shorter leg is equal to  difference between  preceding bypassed nombor Fibonacci and  shorter leg of  preceding segi tiga.

first segi tiga in this series has sides of length 5, 4, and 3. Skipping 8,  next segi tiga has sides of length 13, 12 (5&nbsp;+&nbsp;4&nbsp;+&nbsp;3), and 5 (8&nbsp;&minus;&nbsp;3). Skipping 21,  next segi tiga has sides of length 34, 30 (13&nbsp;+&nbsp;12&nbsp;+&nbsp;5), and 16 (21&nbsp;&minus;&nbsp;5). This series continues indefinitely.  segi tiga 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 segi tiga sides apabila <math>n > 2</math>.

Any four consecutive nombor Fibonacci ''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⏎
Contoh 1: let nombor Fibonacci 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⏎
Contoh 2: let nombor Fibonacci 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>

==Magnitud nombor Fibonacci==
Memandangkan<math>F_n</math> adalah [[berasimptot]] kepada <math>\varphi^n/\sqrt5</math>, bilangan digit dalam asas perwakilan ''b'' <math>F_n\,</math> adalah berasimptot kepada <math>n\,\log_b\varphi</math>.

Dalam asas 10, untuk setiap integer yang lebih besar daripada 1 terdapat 4 atau 5 nombor Fibonacci dengan bilangan digit itu, dalam kebanyakan kes 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>

The Fibonacci number series is used for optional [[lossy compression]] in the [[Interchange_File_Format|IFF]] [[8SVX]] audio file format used on [[Amiga]] computers. The number series  [[companding|compands]] the original audio wave similar to logarithmic methods e.g. [[µ-law]].<ref>Amiga ROM Kernel Reference Manual, Addison-Wesley 1991</ref><ref>[http://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression IFF - MultimediaWiki]</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&ndash;274 | url=http://www.math.ntnu.no/~jarlet/Douady96.pdf  | doi = 10.1006/jtbi.1996.0026 | volume=178|format=PDF}}</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 Computer Science for Fun - cs4fn: Marks for the da Vinci Code: B<!-- Bot generated title -->]</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 iNombor Fibonacci are important dalam run-time analysis of [[algoritma Euclid]] to determine  [[greatest common divisor]] of two integers:  worst case input for this algoritma is a pair of consecutive nombor Fibonacci.

[[Yuri Matiyasevich]] dapat menunjukkan bahawa nombor Fibonacci boleh ditakrifkan daripada [[persamaan Diophantine]], which led to [[Matiyasevich's theorem|his original solution]] of [[Hilbert's tenth problem]].

Nombor Fibonacci occur dalam sums of "shallow" diagonals in [[segi tiga Pascal]] dan [[Lozanić's triangle|segi tiga Lozanić]] (''see "[[Binomial coefficient]]"''). (They occur more obviously in [[segi tiga Hosoya]]).

Setiap integer positif boleh ditulis dalam cara khusus sebagai hasil pertambahan ''lebih dari satu'' nombor Fibonacci yang tersendiri di mana hasil pertambahan tersebut tidak termasuk sebarang daripada dua nombor Fibonacci yang berturutan. This is known as [[Zeckendorf's theorem]], and a sum of nombor Fibonacci that satisfies these conditions is called a Zeckendorf representation.

Prinsip dan nombor Fibonacci juga digunakan dalam [[pasaran perniagaan]] di mana ia digunakan dalam algoritma, aplikasi dan strategi perdagangan. Some typical forms termasuk:  Fibonacci fan, Fibonacci Arc, Fibonacci Retracement and  Fibonacci Time Extension.

Nombor Fibonacci digunakan by some [[pseudorandom number generators]].<!-- Knuth vol. 2 -->

Nombor Fibonacci digunakan in a polyphase version of  [[merge sort]] algoritma in which an unsorted list is divided into two lists whose lengths correspond to sequential nombor Fibonacci - by dividing  list so that  two parts mempunyailengths dalam approximate proportion φ. A tape-drive implementation of  polyphase merge sort was described dalam ''[[The Art of Computer Programming|Art of Computer Programming]]''.

nombor Fibonacci arise dalam analysis of  [[Fibonacci heap]] data structure.

A one-dimensional optimization method, called  [[Fibonacci search technique]], uses nombor Fibonacci.<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>

Siri nombor Fibonacci series is used for optional [[lossy compression]] dalam [[Interchange_File_Format|IFF]] [[8SVX]] audio file format used on [[Amiga]] computers.  number series  [[companding|compands]]  original audio wave similar to logarithmic methods e.g. [[hukum µ]].<ref>Amiga ROM Kernel Reference Manual, Addison-Wesley 1991</ref><ref>[http://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression IFF - MultimediaWiki]</ref>

In [[muzik]], nombor Fibonacci kadangkalanya digunakan to determine tunings, and, as in visual art, to determine  length or size of [[content]] or [[form (music)|formal]] elements. It is commonly thought that  first movement of [[Béla Bartók]]'s ''[[Music for Strings, Percussion, and Celesta]]'' was structured using nombor Fibonacci.

Memandangkan faktor [[pertukaran unit]] [[Batu (ukuran)|batu]] kepada kilometer iaitu 1.609344 hampir dengan [[nisbah keemasan]] (denoted φ),  decomposition of distance in miles into a sum of nombor Fibonacci becomes nearly  kilometer sum apabila nombor Fibonacci are replaced by their successors. This method amounts to a [[radix]] 2 [[Fibonacci coding|number]] [[processor register|register]] in [[asas nisbah keemasan]] φ being shifted. To convert from kilometers to miles, shift  register down  jujukan Fibonacci 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>

==Nombor Fibonacci dalam persekitaran==
[[Image:Helianthus whorl.jpg|thumb|[[Sunflower]] head displaying florets in lingkaran of 34 and 55 around  outside]]
Jujukan Fibonacci muncul dalam persekitaran biologi,<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&ndash;274 | url=http://www.math.ntnu.no/~jarlet/Douady96.pdf  | doi = 10.1006/jtbi.1996.0026 | volume=178|format=PDF}}</ref> in two consecutive nombor Fibonacci, such as branching in trees, arrangement of [[leaves]] on a stem,  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>  flowering of [[artichoke]], an uncurling fern and  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 nombor Fibonacci or [[golden section]]s in nature are found in popular sources, e.g. relating to  breeding of rabbits, lingkaran cengkerang, dan curve of waves{{Fact|date=February 2007}}.  Nombor Fibonacci juga didapati dalam family tree of honeybees. <ref>[http://www.cs4fn.org/maths/bee-davinci.php Computer Science for Fun - cs4fn: Marks for the da Vinci Code: B<!-- Bot generated title -->]</ref>

[[Przemyslaw Prusinkiewicz]] advanced  idea that real instances can be in part understood as  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>

Suatu model corak [[floret]] dalam kepala [[bunga matahari]] diusulkan H. Vogel pada tahun 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 Model ini mempunyai bentuk:
:<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
  | year= 1990
  | location =
  | pages =101-107
  | url =http://algorithmicbotany.org/papers/#webdocs
  | doi =
  | id = ISBN 978-0387972978 }}</ref>
* In 1991, Jean-Claude Perez proposed a connection between [[DNA]] [[base sequence]]s within [[gene sequence]]s and Fibonacci Numbers <ref>J.C. Perez (1991), [http://golden-ratio-in-dna.blogspot.com/2008/01/1991-first-publication-related-to.html "Chaos DNA and Neuro-computers: A Golden Link"], in ''Speculations in Science and Technology'' vol. 14 no. 4, {{ISSN|0155-7785}}</ref>.

== Budaya popular ==
{{main|Nombor Fibonacci dalam budaya popular}}
<!--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>

In 1990, Jean-claude Perez published strong links between [[fractals]] world and Fibonacci numbers index number of  floret and ''c'' is a constant scaling factor;  florets thus lie on [[Fermat's spiral]].  divergence angle, approximately 137.51°, is  [[golden angle]], dividing  circle dalam [[nisbah keemasan]].  Because this ratio is irrational, no floret mempunyai neighbor at exactly  same angle from  center, so  florets pack efficiently.  Because  rational approximations to  nisbah keemasan are of  form F(j):F(j+1),  nearest neighbors of floret number ''n'' are those at ''n''±F(j) for some index ''j'' which depends on ''r'',  distance from  center.  It is often said that sunflowers and similar arrangements mempunyai 55 lingkaran dalam satu arah dan 89 dalam satu arah yang lain (or some other pair of adjacent nombor Fibonacci), but this is benar only of one range of radii, typically  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
  | year= 1990
  | location =
  | pages =101-107
  | url =http://algorithmicbotany.org/papers/#webdocs
  | doi =
  | id = ISBN 978-0387972978 }}</ref>
* Pada 1991, Jean-Claude Perez mengusulkan hubungan natara urutan bes [[DNA]] dalam [[susunan gen]] dengan nombor Fibonacci <ref>J.C. Perez (1991), [http://golden-ratio-in-dna.blogspot.com/2008/01/1991-first-publication-related-to.html "Chaos DNA and Neuro-computers: A Golden Link"], in ''Speculations in Science and Technology'' vol. 14 no. 4, {{ISSN|0155-7785}}</ref>.

==Generalizations==
{{main|Generalizations of Fibonacci numbers}}
Fibonacci sequence has been generalized in many ways. These termasuk:
* Generalizing  index to negative integers to produce  [[Negafibonacci|Nega]]<nowiki/>nombor Fibonacci.
* Generalizing  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. [[nombor Lucas]] 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>. [[jujukan Primefree]] menggunakan 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  sum) of  2 preceding numbers. [[Nomber Pell]] mempunyai ''P<sub>n</sub>'' = 2''P''<sub>''n'' – 1</sub> + ''P''<sub>''n'' – 2</sub>.
* Not adding  immediately preceding numbers.  [[Padovan sequence]] and [[nombor Perrin]] mempunyai P(n) = P(n – 2) + P(n – 3).
* Generating  next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more.
* Adding other objects than integers, misalnya functions or strings -- one essential example is [[polinomial Fibonacci]].

==Numbers properties==
===Periodicity mod ''n'': Pisano periods===
It is easily seen that if  members of  jujukan Fibonacci are taken mod ''n'',  resulting sequence must be [[periodic sequence|periodic]] with period at most <math>n^2</math>.   lengths of  periods for various ''n'' form  so-called [[Pisano period]]s {{OEIS|id=A001175}}.  Determining  Pisano periods in general is an open problem,{{Fact|date=March 2008}} although untuk sebarang particular ''n'' it can be solved as an instance of [[cycle detection]].

==bee ancestry code==
nombor Fibonacci also appear dalam description of  reproduction of a population of idealized bees, according to  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 manakala a female bee will have two.

If one traces 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).   female had two parents, a male and a female, and  male had one female (3 bees).  Those two females each had two parents, and  male had one (5 bees).  This sequence of numbers of parents is jujukan Fibonacci.<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  lineage of distinct parents.

==Miscellaneous==
Pada tahun 1963, John H. E. Cohn proved that  only squares among  nombor Fibonacci 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>

Pada tahun 1990, Jean-claude Perez published strong links between [[fractals]] world and nombor Fibonacci sensitivity <ref>[[IEEE]] [http://ieeexplore.ieee.org/Xplore/login.jsp?url=/iel2/148/3745/00137678.pdf?arnumber=137678 Integers neural network systems (INNS) using resonance propertiesof a Fibonacciapos;s chaotic `golden neuronapos] 1990</ref><ref>[http://golden-ratio-in-dna.blogspot.com/2008/01/1992-order-and-chaos-in-dnathe-denis.html Golden ratio and numbers in DNA] 2008</ref>

==Lihat pula==
*[[Logarithmic spiral]]
*[[b:Atur cara bilangan Fibonacci|Atur cara bilangan Fibonacci]] di [[Bukuwiki]]
*[[Pertubuhan Fibonacci]]
*[[Fibonacci Quarterly]] &mdash; sebuah jurnal akademik devoted pada kajian nombor Fibonacci
*Bilangan [[Negafibonacci]]
*[[Bilangan Lucas]]

==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).
* Hrant Arakelian. ''Mathematics and History of the Golden Section''<nowiki>, Logos 2014, 404 p. ISBN 978-5-98704-663-0 (rus.)</nowiki>
* wallstreetcosmos.com, ''[http://www.wallstreetcosmos.com/elliot.html Fibonacci numbers ''[http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html Golden Section: Phi]'', (2005).
* Ron Knott, ''[http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrep.html Representations of Integers using nombor Fibonacci]'', (2004).
* Hrant Arakelian. ''Mathematics and History of The Golden Section''<nowiki>, Logos 2014, 404 p. ISBN 978-5-98704-663-0 (rus.)</nowiki>
* wallstreetcosmos.com, ''[http://www.wallstreetcosmos.com/elliot.html Fibonacci 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 numbersnumber Information]'', (undated, 2005 or earlier).
* [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html nombor Fibonacci and  Golden Section] – Ron Knott's Surrey University multimedia web site on the Fibonacci numbers, Golden section and  Golden string.
*[http://www.mscs.dal.ca/Fibonacci/ Fibonacci Association] incorporated in [[1963]], focuses on nombor Fibonacci 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 spiralslingkaran in nature]
*[http://web.archive.org/web/20070715032716/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=630&bodyId=1002 Fibonacci Nnumbers] at [http://web.archive.org/web/20060212072618/http://mathdl.maa.org/convergence/1/ Convergence]
* [http://www.tools4noobs.com/online_tools/fibonacci/ Online Fibonacci calculator]

[[Kategori:Fibonacci numbers|*]]
[[Kategori:Articles containing proofs]]

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