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In Mathematics , the Fibonacci numbers form a Sequence defined by the following Recurrence Relation :
:
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}

That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers , also denoted as ''Fn'', for ''n'' = 0, 1, … , are:
: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811…
(Sometimes this sequence is considered to start at ''F''1 = 1, but in this article it is regarded as beginning with ''F''0=0.)

The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci , although they had been described earlier in India .Parmanand Singh. Acharya Hemachandra and the (so called) Fibonacci Numbers. Math . Ed. Siwan , 20(1):28-30,1986.ISSN 0047-6269]Parmanand Singh,"The So-called Fibonacci numbers in ancient and medieval India. Historia Mathematica v12 n3, 229–244,1985


ORIGINS

The Fibonacci numbers first appeared, under the name ''mātrāmeru'' (mountain of cadence), in the work of the Sanskrit Grammarian Pingala (''Chandah-shāstra'', the Art of Prosody, 450 or 200 BC ). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian Mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of Metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c. 1150 ) composed a well known text on these. A commentary on Virahanka by Gopāla in the 12th century also revisits the problem in some detail.

Sanskrit vowel sounds can be long (L) or short (S), and Virahanka's analysis, which came to be known as ''mātrā-vṛtta'', wishes to compute how many metres (''mātrā''s) of a given overall length can be composed of these syllables. If the long syllable is twice as long as the short, the solutions are:
: 1 Mora : 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 series. for items of lengths 1 and 2.

In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci , in his Liber Abaci ( 1202 )1 Chapter II.12, pp. 404–405.. He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that:
  • in the first month there is just one newly-born pair,

  • new-born pairs become fertile from their second month on

  • each month every fertile pair begets a new pair, and

  • the rabbits never die




Closed form expression

Like every sequence defined by linear Recurrence , the Fibonacci numbers have a Closed-form Solution . It has become known as Binet 's formula, even though it was already known by Abraham De Moivre :
:F\left(n ight) = \, , where arphi is the golden ratio.

The Fibonacci recursion

:F(n+2)-F(n+1)-F(n)=0\,

is similar to the defining equation of the golden ratio in the form

:x^2-x-1=0,\,

which is also known as the generating polynomial of the recursion.

Proof (by Induction ):

Any root of the equation above satisfies \begin{matrix}x^2=x+1,\end{matrix}\, and multiplying by x^{n-1}\, shows:
:x^{n+1} = x^n + x^{n-1}\,

By definition arphi is a root of the equation, and the other root is 1- arphi\, .. Therefore:
: arphi^{n+1} = arphi^n + arphi^{n-1}\,

and
:(1- arphi)^{n+1} = (1- arphi)^n + (1- arphi)^{n-1}\, .

Now consider the functions:
:F_{a,b}(n) = a arphi^n+b(1- arphi)^n defined for any real a,b\, .

All these functions satisfy the Fibonacci recursion
:\begin{align}
F_{a,b}(n+1) &= a arphi^{n+1}+b(1- arphi)^{n+1} \
&=a( arphi^{n}+ arphi^{n-1})+b((1- arphi)^{n}+(1- arphi)^{n-1}) \
&=a{ arphi^{n}+b(1- arphi)^{n}}+a{ arphi^{n-1}+b(1- arphi)^{n-1}} \
&=F_{a,b}(n)+F_{a,b}(n-1)
\end{align}
Selecting a=1/\sqrt 5 and b=-1/\sqrt 5 gives the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore:
:F_{a,b}(0)= rac{1}{\sqrt 5}- rac{1}{\sqrt 5}=0\,\!

and
:F_{a,b}(1)= rac{ arphi}{\sqrt 5}- rac{(1- arphi)}{\sqrt 5}= rac{-1+2 arphi}{\sqrt 5}= rac{-1+(1+\sqrt 5)}{\sqrt 5}=1,

establishing the base cases of the induction, proving that
:F(n)= for all n\, .

For any two starting values, a combination a,b can be found such that the function F_{a,b}(n)\, is the exact closed formula for the series.

&= \lim_{n o\infty} rac{a arphi-b(1- arphi)( rac{1- arphi}{ arphi})^n}{a-b( rac{1- arphi}{ arphi})^n} \
&= arphi
\end{align}

or
: ec F_{k+1} = A ec F_{k}.\,

The Eigenvalue s of the matrix A are arphi\,\! and (1- arphi)\,\!, and the elements of the Eigenvector s of A, { arphi \choose 1} and {1 \choose - arphi}, are in the ratios arphi\,\! and (1- arphi\,\!).

This matrix has a determinant of −1, and thus it is a 2×2 Unimodular Matrix . This property can be understood in terms of the Continued Fraction representation for the golden ratio:
: arphi
=1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\;\;\ddots\,}}} \;.
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for arphi\,\!, and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.

The matrix representation gives the following Closed Expression for the Fibonacci numbers:
:\begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix}^n =
\begin{pmatrix} F_{n+1} & F_n \
F_n & F_{n-1} \end{pmatrix}.


Taking the determinant of both sides of this equation yields Cassini's Identity
: F_{n+1}F_{n-1} - F_n^2 = (-1)^n.\,

Additionally, since A^n A^m=A^{m+n} for any square matrix A, the following identities can be derived:
:{F_n}^2 + {F_{n-1}}^2 = F_{2n-1},\,
:F_{n+1}F_{m} + F_n F_{m-1} = F_{m+n}.\,


RECOGNIZING FIBONACCI NUMBERS


Occasionally, the question may arise whether a positive integer z is a Fibonacci number. Since F(n) is the closest integer to arphi^n/\sqrt{5}, the most straightforward test is the identity
:F\bigg(\bigg\lfloor\log_ arphi(\sqrt{5}z)+ rac{1}{2}\bigg floor\bigg)=z,
which is true If And Only If z is a Fibonacci number.

A slightly more sophisticated test uses the fact that the Convergent s of the Continued Fraction representation of arphi are ratios of successive Fibonacci numbers, that is the inequality


RECIPROCAL SUMS




A one-dimensional optimization method, called the Fibonacci Search Technique , uses Fibonacci numbers.5

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 1.609 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 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

head displaying florets in spirals of 34 and 55 around the outside]]
Fibonacci sequences appear in biological settings,6 such as branching in trees, the spiral of shells, the curve of waves, the fruitlets of a Pineapple ,7 an uncurling fern and the arrangement of a Pine Cone .8 Przemyslaw Prusinkiewicz advanced the idea that these can be in part understood as the expression of certain algebraic constraints on Free Group s, specifically as certain Lindenmayer Grammar s.9

A model for the pattern of Floret s in the head of a Sunflower was proposed by H. Vogel in 1979.
  Last Vogel
  First H
  Title A better way to construct the sunflower head
  Journal Mathematical Biosciences
  Issue 44
  Pages 179–189
  Year 1979


This has the form
: heta = rac{2\pi}{\phi^2} n, r = c \sqrt{n}