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[[File:Fibonacci Squares.svg|thumb|A tiling with [[square]]s whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21]]
The Fibonacci numbers were first described in [[Indian mathematics]] as early as 200
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''[[Fibonacci Quarterly]]''. Applications of Fibonacci numbers include computer algorithms such as the [[Fibonacci search technique]] and the [[Fibonacci heap]] [[data structure]], and [[graph (discrete mathematics)|graphs]] called [[Fibonacci cube]]s used for interconnecting parallel and distributed systems. They also appear [[Patterns in nature#Spirals|in biological settings]], such as branching in trees, [[phyllotaxis|the arrangement of leaves on a stem]], the fruit sprouts of a [[pineapple]], the flowering of an [[artichoke]], and the arrangement of a [[pine cone]]'s bracts, though they do not occur in all species.
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The Fibonacci sequence appears in [[Indian mathematics]], in connection with [[Sanskrit prosody]].<ref name="HistoriaMathematica">{{Citation|first=Parmanand|last=Singh|title= The So-called Fibonacci numbers in ancient and medieval India|journal=Historia Mathematica|volume=12|issue=3|pages=229–244|year=1985|doi = 10.1016/0315-0860(85)90021-7|doi-access=free}}</ref><ref name="knuth-v1">{{Citation|title=The Art of Computer Programming|volume=1|first=Donald|last=Knuth| author-link =Donald Knuth |publisher=Addison Wesley|year=1968|isbn=978-81-7758-754-8|url=https://books.google.com/books?id=MooMkK6ERuYC&pg=PA100|page=100|quote=Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns ... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed) ...}}</ref>{{sfn|Livio|2003|p=197}} In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration {{mvar|m}} units is {{math|''F''<sub>''m''+1</sub>}}.<ref name="Donald Knuth 2006 50">{{Citation|title = The Art of Computer Programming | volume = 4. Generating All Trees – History of Combinatorial Generation | first = Donald | last = Knuth | author-link = Donald Knuth |publisher= Addison–Wesley |year= 2006 | isbn= 978-0-321-33570-8 | page = 50 | url= https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50 | quote = it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ... there are exactly Fm+1 of them. For example the 21 sequences when {{math|1=''m'' = 7}} are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)}}</ref>
Knowledge of the Fibonacci sequence was expressed as early as [[Pingala]] ({{circa}} 450 BC–200 BC). Singh cites Pingala's cryptic formula ''misrau cha'' ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for {{mvar|m}} beats ({{math|''F''<sub>''m''+1</sub>}}) is obtained by adding one [S] to the {{math|''F''<sub>''m''</sub>}} cases and one [L] to the {{math|''F''<sub>''m''−1</sub>}} cases.<ref>{{Citation | last = Agrawala | first = VS | year = 1969 | title = ''Pāṇinikālīna Bhāratavarṣa'' (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan | quote = SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala [1969, 463–76], after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC}}</ref> [[Bharata Muni]] also expresses knowledge of the sequence in the ''[[Natya Shastra]]'' (c. 100 BC–c. 350 AD).
However, the clearest exposition of the sequence arises in the work of [[Virahanka]] (c. 700
<blockquote>Variations of two earlier meters [is the variation] ... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21] ... In this way, the process should be followed in all ''mātrā-vṛttas'' [prosodic combinations].{{efn|"For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier—three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven [[Mora (linguistics)|morae]] [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" <ref>{{Citation|last=Velankar|first=HD|year=1962|title='Vṛttajātisamuccaya' of kavi Virahanka|publisher=Rajasthan Oriental Research Institute|___location=Jodhpur|page=101}}</ref>}}</blockquote>
[[Hemachandra]] (c. 1150) is credited with knowledge of the sequence as well,<ref name=GlobalScience/> writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."{{sfn|Livio|2003|p=197–198}}<ref>{{citation|last1=Shah|first1=Jayant|year=1991|title=A History of Piṅgala's Combinatorics|url=https://web.northeastern.edu/shah/papers/Pingala.pdf|publisher=[[Northeastern University]]|page=41|access-date=
===Europe===
[[File:Liber abbaci magliab f124r.jpg|thumb|upright=1.25|A page of [[Fibonacci]]'s {{lang|la|[[Liber Abaci]]}} from the [[National Central Library (Florence)|Biblioteca Nazionale di Firenze]] showing (in box on right) 13 entries of the Fibonacci sequence:<br /> the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.]]
The Fibonacci sequence first appears in the book {{lang|la|[[Liber Abaci]]}} (''The Book of Calculation'', 1202) by [[Fibonacci]],{{Sfn|Sigler|2002|pp=404–405}}<ref>{{citation|url=https://www.math.utah.edu/~beebe/software/java/fibonacci/liber-abaci.html|title=Fibonacci's Liber Abaci (Book of Calculation)|date=13 December 2009|website=[[The University of Utah]]|access-date=
| last = Tassone | first = Ann Dominic
| date = April 1967
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* At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.
At the end of the {{mvar|n}}-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month {{math|''n'' – 2}}) plus the number of pairs alive last month (month {{math|''n'' – 1}}).
| title = Fibonacci's Rabbits | url=http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits | publisher =[[University of Surrey]] Faculty of Engineering and Physical Sciences}}</ref>
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<math display=block>F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5}\right\rceil,\ n \geq 0.</math>
In fact, the rounding error quickly becomes very small as {{mvar|n}} grows, being less than 0.1 for {{math|''n'' ≥ 4}}, and less than 0.01 for {{math|''n'' ≥ 8}}.
<math display=block>n(F) = \left\lfloor \log_\varphi \sqrt{5}F\right\rceil,\ F \geq 1.</math>
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=== Limit of consecutive quotients ===
[[Johannes Kepler]] observed that the ratio of consecutive Fibonacci numbers [[convergent sequence|converges]]. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio
<math display=block>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>
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=== Identification ===
Binet's formula provides a proof that a positive integer {{mvar|x}} is a Fibonacci number [[if and only if]] at least one of <math>5x^2+4</math> or <math>5x^2-4</math> is a [[Square number|perfect square]].<ref>{{Citation | title = Fibonacci is a Square | last1 = Gessel | first1 = Ira | journal = [[The Fibonacci Quarterly]] | volume = 10 | issue = 4 | pages = 417–19 |date=October 1972 | url = https://www.fq.math.ca/Scanned/10-4/advanced10-4.pdf | access-date =
<math display=block>\varphi^n = \frac{F_n\sqrt{5} \pm \sqrt{5{F_n}^{\!2} + 4(-1)^n}}{2}.</math>
Comparing this to <math>\varphi^n = F_n \varphi + F_{n-1} = (F_n\sqrt{5} + F_n + 2 F_{n-1})/2</math>, it follows that
: <math display=block>5{F_n}^{\!2} + 4(-1)^n = (F_n + 2F_{n-1})^2\,.</math>
In particular, the left-hand side is a perfect square.
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<math display=block> \vec F_{k+1} = \mathbf{A} \vec F_{k},</math>
which yields <math>\vec F_n = \mathbf{A}^n \vec F_0</math>.
<math display=block>\vec \mu=\begin{pmatrix} \varphi \\ 1 \end{pmatrix}, \quad \vec\nu=\begin{pmatrix} -\varphi^{-1} \\ 1 \end{pmatrix}.</math>
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\end{align}</math>
where
<math display=block>
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<math display=block>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}.</math>
The [[convergent (continued fraction)|convergents]] of the continued fraction for {{mvar|φ}} are ratios of successive Fibonacci numbers: {{math|1=''φ''<sub>''n''</sub> = ''F''<sub>''n''+1</sub> / ''F''<sub>''n''</sub>}} is the {{mvar|n}}-th convergent, and the {{math|(''n'' + 1)}}-st convergent can be found from the recurrence relation {{math|1=''φ''<sub>''n''+1</sub> = 1 + 1 / ''φ''<sub>''n''</sub>}}.<ref>{{Cite web |title=The Golden Ratio, Fibonacci Numbers and Continued Fractions. |url=https://nrich.maths.org/2737 |access-date=2024-03-22 |website=nrich.maths.org |language=en}}</ref> The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
<math display=block>\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}.</math>
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Following the same logic as before, by summing the cardinality of each set we see that
: <math>F_{n+2} = F_n + F_{n-1} + ... + |\{(1,1,...,1,2)\}| + |\{(1,1,...,1)\}|</math>
... where the last two terms have the value <math>F_1 = 1</math>.
A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:
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A different trick may be used to prove
<math display=block>\sum_{i=1}^n F_i^2 = F_n F_{n+1}</math>
or in words, the sum of the squares of the first Fibonacci numbers up to <math>F_n</math> is the product of the {{mvar|n}}-th and {{math|(''n'' + 1)}}-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size <math>F_n \times F_{n+1}</math> and decompose it into squares of size <math>F_n, F_{n-1}, ..., F_1</math>; from this the identity follows by comparing
[[File:Fibonacci Squares.svg|frameless|260x260px]]
=== Induction proofs ===
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<math display=block>
s(z) = \sum_{k=0}^\infty F_k z^k = 0 + z + z^2 + 2z^3 + 3z^4 + 5z^5 + \
</math>
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<math display=block>s(z) = s\!\left(-\frac{1}{z}\right).</math>
Using <math>z</math> equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of <math>s(z)</math>.
== Reciprocal sums ==
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Every third number of the sequence is even (a multiple of <math>F_3=2</math>) and, more generally, every {{mvar|k}}-th number of the sequence is a multiple of ''F<sub>k</sub>''. Thus the Fibonacci sequence is an example of a [[divisibility sequence]]. In fact, the Fibonacci sequence satisfies the stronger divisibility property<ref>{{Citation | first = Paulo | last = Ribenboim | author-link = Paulo Ribenboim | title = My Numbers, My Friends | publisher = Springer-Verlag | year = 2000}}</ref><ref>{{Citation | last1 = Su | first1 = Francis E | others = et al | publisher = HMC | url = http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml | contribution = Fibonacci GCD's, please | year = 2000 | title = Mudd Math Fun Facts | access-date = 2007-02-23 | archive-url = https://web.archive.org/web/20091214092739/http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml | archive-date = 2009-12-14 | url-status = dead }}</ref>
<math display=block>\gcd(F_a,F_b,F_c,\ldots) = F_{\gcd(a,b,c,\ldots)}\,</math>
where {{math|gcd}} is the [[greatest common divisor]] function.
In particular, any three consecutive Fibonacci numbers are pairwise [[Coprime integers|coprime]] because both <math>F_1=1</math> and <math>F_2 = 1</math>.
: <math>\gcd(F_n, F_{n+1}) = \gcd(F_n, F_{n+2}) = \gcd(F_{n+1}, F_{n+2}) = 1</math>
for every {{mvar|n}}.
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The above formula can be used as a [[primality test]] in the sense that if
<math display=block>n \mid F_{n \;-\, \left(\frac{5}{n}\right)},</math>
where the Legendre symbol has been replaced by the [[Jacobi symbol]], then this is evidence that {{mvar|n}} is a prime, and if it fails to hold, then {{mvar|n}} is definitely not a prime. If {{mvar|n}} is [[composite number|composite]] and satisfies the formula, then {{mvar|n}} is a ''Fibonacci pseudoprime''.
<math display=block> \begin{pmatrix} F_{m+1} & F_m \\ F_m & F_{m-1} \end{pmatrix} \equiv \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^m \pmod n.</math>
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<math display=block>F_1 = 1,\ F_3 = 2,\ F_5 = 5,\ F_7 = 13,\ F_9 = {\color{Red}34} = 2 \cdot 17,\ F_{11} = 89,\ F_{13} = 233,\ F_{15} = {\color{Red}610} = 2 \cdot 5 \cdot 61.</math>
All known factors of Fibonacci numbers {{math|''F''(''i'')}} for all {{math|''i'' < 50000}} are collected at the relevant repositories.<ref>{{Citation | url = https://mersennus.net/fibonacci/ | title = Fibonacci and Lucas factorizations | publisher = Mersennus}} collects all known factors of {{math|''F''(''i'')}} with {{math|''i'' < 10000}}
=== Periodicity modulo ''n'' ===
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* Generalizing the index to [[real number]]s using a modification of Binet's formula.<ref name="MathWorld" />
* Starting with other integers. [[Lucas number]]s have {{math|1=''L''<sub>1</sub> = 1}}, {{math|1=''L''<sub>2</sub> = 3}}, and {{math|1=''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 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 number]]s have {{math|1=''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 {{math|1=''P''(''n'') = ''P''(''n'' − 2) + ''P''(''n'' − 3)}}.
* Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as ''n-Step Fibonacci numbers''.<ref>{{citation
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The Fibonacci numbers can be found in different ways among the set of [[binary numeral system|binary]] [[String (computer science)|strings]], or equivalently, among the [[subset]]s of a given set.
* The number of binary strings of length {{mvar|n}} without consecutive {{math|1}}s is the Fibonacci number {{math|''F''<sub>''n''+2</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''<sub>6</sub> = 8}} without consecutive {{math|1}}s—they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010.
* The number of binary strings of length {{mvar|n}} without an odd number of consecutive {{math|1}}s is the Fibonacci number {{math|''F''<sub>''n''+1</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''<sub>5</sub> = 5}} without an odd number of consecutive {{math|1}}s—they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets {{mvar|S}} of {{math|{{mset|1, ..., ''n''}}}} without an odd number of consecutive integers is {{math|''F''<sub>''n''+1</sub>}}. A bijection with the sums to {{mvar|n}} is to replace 1 with 0 and 2 with 11.
* The number of binary strings of length {{mvar|n}} without an even number of consecutive {{math|0}}s or {{math|1}}s is {{math|2''F''<sub>''n''</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=2''F''<sub>4</sub> = 6}} without an even number of consecutive {{math|0}}s or {{math|1}}s—they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
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* A one-dimensional optimization method, called the [[Fibonacci search technique]], uses Fibonacci numbers.<ref>{{Citation| first1 = M | last1 = Avriel | first2 = DJ | last2 = Wilde | title= Optimality of the Symmetric Fibonacci Search Technique |journal=Fibonacci Quarterly|year=1966 |issue=3 |pages= 265–69| doi = 10.1080/00150517.1966.12431364 }}</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 such as [[μ-law]].<ref>{{Citation | title = Amiga ROM Kernel Reference Manual | publisher = Addison–Wesley | year = 1991}}</ref><ref>{{Citation | url = https://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression | contribution = IFF | title = Multimedia Wiki}}</ref>
* Some Agile teams use a modified series called the "Modified Fibonacci Series" in [[planning poker]], as an estimation tool. Planning Poker is a formal part of the [[Scaled agile framework|Scaled Agile Framework]].<ref>{{citation|author=Dean Leffingwell |url=https://www.scaledagileframework.com/story/ |title=Story |publisher=Scaled Agile Framework |date=2021-07-01 |
* [[Fibonacci coding]]
* [[Negafibonacci coding]]
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[[File:FibonacciChamomile.PNG|thumb|[[Yellow chamomile]] head showing the arrangement in 21 (blue) and 13 (cyan) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.]]
Fibonacci sequences appear in biological settings,<ref>{{Citation |first1=S |last1=Douady |first2=Y |last2=Couder |title=Phyllotaxis as a Dynamical Self Organizing Process |journal=Journal of Theoretical Biology |year=1996 |issue=3 |pages=255–74 |url=http://www.math.ntnu.no/~jarlet/Douady96.pdf |doi=10.1006/jtbi.1996.0026 |volume=178 |url-status=dead |archive-url=https://web.archive.org/web/20060526054108/http://www.math.ntnu.no/~jarlet/Douady96.pdf |archive-date=2006-05-26 }}</ref> such as branching in trees, [[Phyllotaxis|arrangement of leaves on a stem]], the fruitlets of a [[pineapple]],<ref>{{Citation | first1=Judy |last1=Jones | first2=William | last2=Wilson |title=An Incomplete Education |publisher=Ballantine Books |year=2006 |isbn=978-0-7394-7582-9 |page=544 |chapter=Science}}</ref> the flowering of [[artichoke]], the arrangement of a [[pine cone]],<ref>{{Citation| first=A | last=Brousseau |title=Fibonacci Statistics in Conifers | journal=[[Fibonacci Quarterly]] |year=1969 |issue=7 |pages=525–32}}</ref> and the family tree of [[honeybee]]s.<ref>{{citation|url = https://www.cs4fn.org/maths/bee-davinci.php |work = Maths | publisher = Computer Science For Fun: CS4FN |title = Marks for the da Vinci Code: B–}}</ref><ref>{{Citation|first1=T.C.|last1=Scott|first2=P.|last2=Marketos| url = http://www-history.mcs.st-andrews.ac.uk/Publications/fibonacci.pdf | title = On the Origin of the Fibonacci Sequence | publisher = [[MacTutor History of Mathematics archive]], University of St Andrews| date = March 2014}}</ref> [[Kepler]] pointed out the presence of the Fibonacci sequence in nature, using it to explain the ([[golden ratio]]-related) [[pentagon]]al form of some flowers.{{sfn|Livio|2003|p=110}} Field [[Leucanthemum vulgare|daisies]] most often have petals in counts of Fibonacci numbers.{{sfn|Livio|2003|pp=112–13}} In 1830, [[Karl Friedrich Schimper]] and [[Alexander Braun]] discovered that the [[Parastichy|parastichies]] (spiral [[phyllotaxis]]) of plants were frequently expressed as fractions involving Fibonacci numbers.<ref>{{Citation |first =Franck |last = Varenne |title = Formaliser le vivant - Lois, Théories, Modèles |
[[Przemysław Prusinkiewicz]] advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on [[free group]]s, specifically as certain [[L-system|Lindenmayer grammars]].<ref>{{Citation|first1 = Przemyslaw |last1 = Prusinkiewicz | first2 = James | last2 = Hanan| title = Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics) |publisher= [[Springer Science+Business Media|Springer-Verlag]] |year=1989 |isbn=978-0-387-97092-9}}</ref>
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<math display=block>\theta = \frac{2\pi}{\varphi^2} n,\ r = c \sqrt{n}</math>
where {{mvar|n}} is the index number of the floret and {{mvar|c}} is a constant scaling factor; the florets thus lie on [[Fermat's spiral]]. The divergence [[angle]], approximately 137.51°, is the [[golden angle]], dividing the circle in the golden ratio.
Fibonacci numbers also appear in the ancestral pedigrees of [[bee]]s (which are [[haplodiploid]]s), according to the following rules:
Line 772 ⟶ 767:
It has similarly been noticed that the number of possible ancestors on the human [[X chromosome]] inheritance line at a given ancestral generation also follows the Fibonacci sequence.<ref name="xcs">{{citation|last=Hutchison|first=Luke|date=September 2004|title=Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships|url=http://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|journal=Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04)|access-date=2016-09-03|archive-date=2020-09-25|archive-url=https://web.archive.org/web/20200925132536/https://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|url-status=dead}}</ref> A male individual has an X chromosome, which he received from his mother, and a [[Y chromosome]], which he received from his father. The male counts as the "origin" of his own X chromosome (<math>F_1=1</math>), and at his parents' generation, his X chromosome came from a single parent {{nowrap|(<math>F_2=1</math>)}}. The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_3=2</math>)}}. The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_4=3</math>)}}. Five great-great-grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_5=5</math>)}}, etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a [[Founder effect|population founder]] appears on all lines of the genealogy.)
===Other===
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* [[Mario Merz]] included the Fibonacci sequence in some of his artworks beginning in 1970.{{sfn|Livio|2003|p=176}}
* [[Joseph Schillinger]] (1895–1943) developed [[Schillinger System|a system of composition]] which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.{{sfn|Livio|2003|p=193}} See also {{slink|Golden ratio|Music}}.
* In [[software development]], Fibonacci numbers are often used by [[Agile management|agile]] teams operating under the [[Scrum (software development)|Scrum]] framework to size their [[product backlog]] items.<ref>{{cite web |last1=Kathuria |first1=Madhur |title=A Guide to Using the Fibonacci Sequence in Scrum |url=https://resources.scrumalliance.org/Article/guide-using-fibonacci-sequence-scrum |website=Scrum Alliance |access-date=8 August 2025}}</ref>
== See also ==
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| last1 =Borwein
| first1 =Jonathan M.
|
|
| title =Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
| pages =91–101
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| and link back to that category using the {{dmoz}} template. |
======================= {{No more links}} =====================-->
* {{YouTube|id=hbUQlrLDAgw|title=Fibonacci Sequence and Golden Ratio: Mathematics in the Modern World - Mathuklasan with Sir Ram}} - animation of sequence, spiral, golden ratio, rabbit pair growth.
* [https://www.mathpages.com/home/kmath078/kmath078.htm Periods of Fibonacci Sequences Mod m] at MathPages
* [http://www.physorg.com/news97227410.html Scientists find clues to the formation of Fibonacci spirals in nature]
| |||