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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, ... (Sometimes this sequence is considered to start at F1 = 1, but in this article it is regarded as beginning with F0=0.)

The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in Indian mathematics.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 grammarians Pingala (Chandah-shāstra, the Art of Prosody, 450 BC or 200 BC). Prosody (linguistics) was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematicians Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of Vedic meter with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well known text on these. A commentary on Virahanka's work by Gopala (mathematician) 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 (mātrās) of a given overall length can be composed of these syllables. If the long syllable is twice as long as the short, the solutions are: 1 mora (linguistics): 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. Donald Knuth reviews this work in The Art of Computer Programming as equivalent formulations of the bin packing problem for items of lengths 1 and 2.

In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202) Chapter II.12, pp. 404–405.. He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that:

Let the population at month n be F(n). At this time, only rabbits who were alive at month n−2 are fertile and produce offspring, so F(n−2) pairs are added to the current population of F(n−1). Thus the total is F(n) = F(n−1) + F(n−2).{{cite web | last = Knott | first = Ron | title = Fibonacci's Rabbits | url=http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits | publisher =[University of Surrey School of Electronics and Physical Sciences-->

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:

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.

Relation to the golden ratio Golden ratio defined The golden ratio \varphi (phi (letter)), also written \tau (tau), is defined as the ratio that results when a line is divided so that the whole line has the same ratio to the larger segment as the larger segment has to the smaller segment. Expressed algebraically, normalising the larger part to unit length, it is the positive solution of the equation: \frac{x}{1}=\frac{1}{x-1} or equivalently x^2-x-1=0,\,

which is equal to: \varphi = \frac{1 + \sqrt{5-->{2} = 0.5 + \sqrt{1.25} \approx 1.618\,033\,988\,749\,894\,848\,204\,586\,834\,366\,.

Closed form expression Like every sequence defined by linear Recurrence relation, the Fibonacci numbers have a closed-form expression. It has become known as Jacques Philippe Marie Binet's formula, even though it was already known by Abraham de Moivre: F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5-->\, , where \varphi 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 Mathematical 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 \varphi is a root of the equation, and the other root is 1-\varphi\, .. Therefore: \varphi^{n+1} = \varphi^n + \varphi^{n-1}\,

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

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

All these functions satisfy the Fibonacci recursion \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}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)=\frac{1}{\sqrt 5}-\frac{1}{\sqrt 5}=0\,\!

and 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,

establishing the base cases of the induction, proving that F(n)={{\varphi^n-(1-\varphi)^n} \over {\sqrt 5--> 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.

Since \begin{matrix}|1-\varphi|^n/\sqrt 5 < 1/2\end{matrix} for all n\geq 0\, , F(n)\, is the closest integer to \varphi^n/\sqrt 5\, .For computational purposes, this is expressed using the floor function: F(n)=\bigg\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\bigg\rfloor.

Limit of consecutive quotients Johannes Kepler pointed out that the ratio of consecutive Fibonacci numbers converges, stating 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 concludes that the limit approaches the golden ratio \varphi Strena seu de Nive Sexangula (1611)

\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi, This convergence does not depend on the starting values chosen, excluding 0, 0.

Proof:

It follows from the explicit formula that for any real a \ne 0, b \ne 0: \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} because \bigl|{\tfrac{1-\varphi}{\varphi-->\bigr| < 1 and thus \lim_{n\to\infty}\left(\tfrac{1-\varphi}{\varphi}\right)^n=0

Matrix form A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is {F_{k+2} \choose F_{k+1--> = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k-->

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

The eigenvalues of the matrix A are \varphi\,\! and (1-\varphi)\,\!, and the elements of the eigenvectors of A, {\varphi \choose 1} and {1 \choose -\varphi}, are in the ratios \varphi\,\! and (1-\varphi\,\!).

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: \varphi =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 \varphi\,\!, 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 \varphi^n/\sqrt{5}, the most straightforward test is the identity F\bigg(\bigg\lfloor\log_\varphi(\sqrt{5}z)+\frac{1}{2}\bigg\rfloor\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 (continued fraction)s of the continued fraction representation of \varphi are ratios of successive Fibonacci numbers, that is the inequality \bigg|\varphi-\frac{p}{q}\bigg|} = \frac {1}{10^{2k + 2} - 10^{k + 1} - 1}

for all integers k >= 0.

Conversely, \sum_{n=0}^\infty\,\frac{F_n}{k^{n-->\,=\,\frac{k}{k^{2}-k-1}

Reciprocal sums It credits some formulae to -->Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as \sum_{k=0}^\infty \frac{1}{F_{2k+1--> = \frac{\sqrt{5-->{4}\vartheta_2^2 \left(0, \frac{3-\sqrt 5}{2}\right) ,

and the sum of squared reciprocal Fibonacci numbers as \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).

If we add 1 to each Fibonacci number in the first sum, there is also the closed form \sum_{k=0}^\infty \frac{1}{1+F_{2k+1--> = \frac{\sqrt{5-->{2},

and there is a nice nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio, \sum_{k=1}^\infty \frac{(-1)^{k+1-->{\sum_{j=1}^k {F_{j-->^2} = \frac{\sqrt{5}-1}{2}.

Results such as these make it plausible that a closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but none is yet known. Despite that, the reciprocal Fibonacci constant \psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots

has been proved irrational number by Richard André-Jeannin.

Primes and divisibility A Fibonacci prime is a Fibonacci number that is prime number . 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 F4 = 3.

Any three consecutive Fibonacci numbers, taken two at a time, are relatively prime: that is, greatest common divisor(Fn,Fn+1) = gcd(Fn,Fn+2) = 1. More generally, gcd(Fn, Fm) = Fgcd(n,m).Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000

A proof of this striking fact is online at Harvey Mudd College's Fun Math site

Right triangles Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.

The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely.

Magnitude of Fibonacci numbers The number of base b digits of F_n\, asymptotes to n\,\log_b\varphi.

Applications The Fibonacci numbers are important in the run-time analysis of Euclidean 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 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.

Fibonacci numbers are used by some pseudorandom number generators.

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 form (music) 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 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 Fibonacci coding processor register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead. An Application of the Fibonacci Number Representation A Practical Use of the Sequence Zeckendorf representation

Fibonacci numbers in nature head displaying florets in spirals of 34 and 55 around the outsideFibonacci sequences appear in biological settings, such as branching in trees, the fruitlets of a pineapple, 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 L-systems.

A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.{{Citation | 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 \theta = \frac{2\pi}{\phi^2} n, 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.{{cite book | last =Prusinkiewicz | first =Przemyslaw | authorlink =Przemyslaw Prusinkiewicz | coauthors =[Aristid Lindenmayer | title =[The Algorithmic Beauty of Plants | publisher =Springer-Verlag | date =1990 | location = | pages =101-107 | url =http://algorithmicbotany.org/papers/#webdocs | doi = | id = ISBN 978-0387972978 -->

Popular culture Because the Fibonacci sequence is easy for non-mathematicians to understand, there are many examples of the Fibonacci numbers being used in popular culture.

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

Numbers Properties Divisibility by 11 Any ten consecutive Fibonacci numbers are divisible by 11. Examples: 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11 * 13 55 + 89 + 144 + 233 + 377 + 610 + 987 + 1597 + 2584 + 4181 = 10857 = 11 * 987 Another interesting fact is that the sum of ten consecutive numbers is always equal to 11 times the 7th number used in the sum.

Periodicity of last n digits One property of the Fibonacci numbers is that the last n digits have the following periodicity: Mathematician Dov Jarden proved that for n greater than 2 the periodicity is 15 * 10n-1.

Sum of all numbers The sum of the first n elements is equal to F(n+2)-1.

Example: S(10) = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 144 - 1

1/89 The 11th Fibonacci number is 89. 1/89 = 0.01123595... Considering the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, ... if they are arranged like bellow 0.01 0.001 0.0002 0.00003 0.000005 0.0000008 0.00000013 0.000000021 ... and then summed, the result is 0.01123595... which is 1/89. Pythagorean triples Any four consecutive Fibonacci numbers Fn, Fn+1, Fn+2 and Fn+3 can be used to generate a Pythagorean triple: 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 \,. Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then: a = 1 \times 5 = 5 b = 2 \times 2 \times 3 = 12 c = 2^2 + 3^2 = 13 5^2 + 12^2 = 13^2 \,. Example 2: let the Fibonacci numbers be 8, 13, 21 and 34. Then: a = 8 \times 34 = 272 b = 2 \times 13 \times 21 = 546 c = 13^2 + 21^2 = 610 272^2 + 546^2 = 610^2 \,.

See also

References

External links



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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, ... (Sometimes this sequence is considered to start at F1 = 1, but in this article it is regarded as beginning with F0=0.)

The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in Indian mathematics.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 grammarians Pingala (Chandah-shāstra, the Art of Prosody, 450 BC or 200 BC). Prosody (linguistics) was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematicians Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of Vedic meter with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well known text on these. A commentary on Virahanka's work by Gopala (mathematician) 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 (mātrās) of a given overall length can be composed of these syllables. If the long syllable is twice as long as the short, the solutions are: 1 mora (linguistics): 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. Donald Knuth reviews this work in The Art of Computer Programming as equivalent formulations of the bin packing problem for items of lengths 1 and 2.

In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202) Chapter II.12, pp. 404–405.. He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that:

Let the population at month n be F(n). At this time, only rabbits who were alive at month n−2 are fertile and produce offspring, so F(n−2) pairs are added to the current population of F(n−1). Thus the total is F(n) = F(n−1) + F(n−2).{{cite web | last = Knott | first = Ron | title = Fibonacci's Rabbits | url=http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits | publisher =[University of Surrey School of Electronics and Physical Sciences-->

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:

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.

Relation to the golden ratio Golden ratio defined The golden ratio \varphi (phi (letter)), also written \tau (tau), is defined as the ratio that results when a line is divided so that the whole line has the same ratio to the larger segment as the larger segment has to the smaller segment. Expressed algebraically, normalising the larger part to unit length, it is the positive solution of the equation: \frac{x}{1}=\frac{1}{x-1} or equivalently x^2-x-1=0,\,

which is equal to: \varphi = \frac{1 + \sqrt{5-->{2} = 0.5 + \sqrt{1.25} \approx 1.618\,033\,988\,749\,894\,848\,204\,586\,834\,366\,.

Closed form expression Like every sequence defined by linear Recurrence relation, the Fibonacci numbers have a closed-form expression. It has become known as Jacques Philippe Marie Binet's formula, even though it was already known by Abraham de Moivre: F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5-->\, , where \varphi 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 Mathematical 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 \varphi is a root of the equation, and the other root is 1-\varphi\, .. Therefore: \varphi^{n+1} = \varphi^n + \varphi^{n-1}\,

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

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

All these functions satisfy the Fibonacci recursion \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}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)=\frac{1}{\sqrt 5}-\frac{1}{\sqrt 5}=0\,\!

and 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,

establishing the base cases of the induction, proving that F(n)={{\varphi^n-(1-\varphi)^n} \over {\sqrt 5--> 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.

Since \begin{matrix}|1-\varphi|^n/\sqrt 5 < 1/2\end{matrix} for all n\geq 0\, , F(n)\, is the closest integer to \varphi^n/\sqrt 5\, .For computational purposes, this is expressed using the floor function: F(n)=\bigg\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\bigg\rfloor.

Limit of consecutive quotients Johannes Kepler pointed out that the ratio of consecutive Fibonacci numbers converges, stating 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 concludes that the limit approaches the golden ratio \varphi Strena seu de Nive Sexangula (1611)

\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi, This convergence does not depend on the starting values chosen, excluding 0, 0.

Proof:

It follows from the explicit formula that for any real a \ne 0, b \ne 0: \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} because \bigl|{\tfrac{1-\varphi}{\varphi-->\bigr| < 1 and thus \lim_{n\to\infty}\left(\tfrac{1-\varphi}{\varphi}\right)^n=0

Matrix form A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is {F_{k+2} \choose F_{k+1--> = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k-->

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

The eigenvalues of the matrix A are \varphi\,\! and (1-\varphi)\,\!, and the elements of the eigenvectors of A, {\varphi \choose 1} and {1 \choose -\varphi}, are in the ratios \varphi\,\! and (1-\varphi\,\!).

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: \varphi =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 \varphi\,\!, 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 \varphi^n/\sqrt{5}, the most straightforward test is the identity F\bigg(\bigg\lfloor\log_\varphi(\sqrt{5}z)+\frac{1}{2}\bigg\rfloor\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 (continued fraction)s of the continued fraction representation of \varphi are ratios of successive Fibonacci numbers, that is the inequality \bigg|\varphi-\frac{p}{q}\bigg|} = \frac {1}{10^{2k + 2} - 10^{k + 1} - 1}

for all integers k >= 0.

Conversely, \sum_{n=0}^\infty\,\frac{F_n}{k^{n-->\,=\,\frac{k}{k^{2}-k-1}

Reciprocal sums It credits some formulae to -->Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as \sum_{k=0}^\infty \frac{1}{F_{2k+1--> = \frac{\sqrt{5-->{4}\vartheta_2^2 \left(0, \frac{3-\sqrt 5}{2}\right) ,

and the sum of squared reciprocal Fibonacci numbers as \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).

If we add 1 to each Fibonacci number in the first sum, there is also the closed form \sum_{k=0}^\infty \frac{1}{1+F_{2k+1--> = \frac{\sqrt{5-->{2},

and there is a nice nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio, \sum_{k=1}^\infty \frac{(-1)^{k+1-->{\sum_{j=1}^k {F_{j-->^2} = \frac{\sqrt{5}-1}{2}.

Results such as these make it plausible that a closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but none is yet known. Despite that, the reciprocal Fibonacci constant \psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots

has been proved irrational number by Richard André-Jeannin.

Primes and divisibility A Fibonacci prime is a Fibonacci number that is prime number . 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 F4 = 3.

Any three consecutive Fibonacci numbers, taken two at a time, are relatively prime: that is, greatest common divisor(Fn,Fn+1) = gcd(Fn,Fn+2) = 1. More generally, gcd(Fn, Fm) = Fgcd(n,m).Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000

A proof of this striking fact is online at Harvey Mudd College's Fun Math site

Right triangles Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.

The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely.

Magnitude of Fibonacci numbers The number of base b digits of F_n\, asymptotes to n\,\log_b\varphi.

Applications The Fibonacci numbers are important in the run-time analysis of Euclidean 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 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.

Fibonacci numbers are used by some pseudorandom number generators.

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 form (music) 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 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 Fibonacci coding processor register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead. An Application of the Fibonacci Number Representation A Practical Use of the Sequence Zeckendorf representation

Fibonacci numbers in nature head displaying florets in spirals of 34 and 55 around the outsideFibonacci sequences appear in biological settings, such as branching in trees, the fruitlets of a pineapple, 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 L-systems.

A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.{{Citation | 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 \theta = \frac{2\pi}{\phi^2} n, 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.{{cite book | last =Prusinkiewicz | first =Przemyslaw | authorlink =Przemyslaw Prusinkiewicz | coauthors =[Aristid Lindenmayer | title =[The Algorithmic Beauty of Plants | publisher =Springer-Verlag | date =1990 | location = | pages =101-107 | url =http://algorithmicbotany.org/papers/#webdocs | doi = | id = ISBN 978-0387972978 -->

Popular culture Because the Fibonacci sequence is easy for non-mathematicians to understand, there are many examples of the Fibonacci numbers being used in popular culture.

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

Numbers Properties Divisibility by 11 Any ten consecutive Fibonacci numbers are divisible by 11. Examples: 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11 * 13 55 + 89 + 144 + 233 + 377 + 610 + 987 + 1597 + 2584 + 4181 = 10857 = 11 * 987 Another interesting fact is that the sum of ten consecutive numbers is always equal to 11 times the 7th number used in the sum.

Periodicity of last n digits One property of the Fibonacci numbers is that the last n digits have the following periodicity: Mathematician Dov Jarden proved that for n greater than 2 the periodicity is 15 * 10n-1.

Sum of all numbers The sum of the first n elements is equal to F(n+2)-1.

Example: S(10) = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 144 - 1

1/89 The 11th Fibonacci number is 89. 1/89 = 0.01123595... Considering the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, ... if they are arranged like bellow 0.01 0.001 0.0002 0.00003 0.000005 0.0000008 0.00000013 0.000000021 ... and then summed, the result is 0.01123595... which is 1/89. Pythagorean triples Any four consecutive Fibonacci numbers Fn, Fn+1, Fn+2 and Fn+3 can be used to generate a Pythagorean triple: 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 \,. Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then: a = 1 \times 5 = 5 b = 2 \times 2 \times 3 = 12 c = 2^2 + 3^2 = 13 5^2 + 12^2 = 13^2 \,. Example 2: let the Fibonacci numbers be 8, 13, 21 and 34. Then: a = 8 \times 34 = 272 b = 2 \times 13 \times 21 = 546 c = 13^2 + 21^2 = 610 272^2 + 546^2 = 610^2 \,.

See also

References

External links



 

Fibonacci Sequence



 
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