Fibonacci numbers
In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci, whose Liber Abaci published in 1202 introduced the sequence to Western European mathematics.
The sequence is defined by the following recurrence relation:
That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as F_{n}, for n = 0, 1, 2, … , 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, ...<ref> By modern convention, the sequence begins with F_{0}=0. The Liber Abici began the sequence with F_{1} = 1, omitting the initial 0, and the sequence is still written this way by some.</ref>
Each third number of the series is an even number.
The sequence named after Fibonacci was first described in Indian mathematics.<ref>Parmanand Singh. "Acharya Hemachandra and the (so called) Fibonacci Numbers". Math. Ed. Siwan, 20(1):2830, 1986. ISSN 00476269]</ref><ref>Parmanand Singh,"The Socalled Fibonacci numbers in ancient and medieval India." Historia Mathematica 12(3), 229–44, 1985.</ref>
The sequence extended to negative index n satisfies F_{n} = F_{n−1} + F_{n−2} for all integers n, and F_{n} = (−1)^{n+1}F_{n}:
.., 8, 5, 3, 2, 1, 1, followed by the sequence above.
Contents
 1 Origins
 2 Relation to the golden ratio
 3 Matrix form
 4 Recognizing Fibonacci numbers
 5 Identities
 6 Power series
 7 Reciprocal sums
 8 Primes and divisibility
 9 Right triangles
 10 Magnitude of Fibonacci numbers
 11 Applications
 12 Fibonacci numbers in nature
 13 Popular culture
 14 Generalizations
 15 Numbers properties
 16 The bee ancestry code
 17 Miscellaneous
 18 See also
 19 References
 20 External links
Origins
The Fibonacci numbers first appeared, under the name mātrāmeru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandahshāstra, 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 wellknown text on these. A commentary on Virahanka's work 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 mātrāvṛtta, 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: 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)<ref>{{
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Relation to the golden ratio
Closed form expression
Like every sequence defined by linear recurrence, the Fibonacci numbers have a closedform solution. It has become known as Binet's formula, even though it was already known by Abraham de Moivre:
 where is the golden ratio (note, that , as can be seen from the defining equation below).
The Fibonacci recursion
is similar to the defining equation of the golden ratio in the form
which is also known as the generating polynomial of the recursion.
Proof by induction
Any root of the equation above satisfies and multiplying by shows:
By definition is a root of the equation, and the other root is . Therefore:
and
Both and 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 twodimensional linear vector space; the original Fibonacci sequence can be found in this space.
Linear combinations of series and , with coefficients a and b, can be defined by
 for any real
All thusdefined series satisfy the Fibonacci recursion
Requiring that and yields and , resulting in the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore, an explicit check can be made:
and
establishing the base cases of the induction, proving that
 for all
Therefore, for any two starting values, a combination can be found such that the function is the exact closed formula for the series.
Computation by rounding
Since for all , the number is the closest integer to Therefore it can be found by rounding, or in terms of the floor function:
Limit of consecutive quotients
Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost”, and concluded that the limit approaches the golden ratio .<ref>{{
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This convergence does not depend on the starting values chosen, excluding 0, 0.
Proof:
It follows from the explicit formula that for any real
because and thus
Decomposition of powers of the golden ratio
Since the golden ratio satisfies the equation
this expression can be used to decompose higher powers as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients, thus closing the loop:
This expression is also true for if the Fibonacci sequence is extended to negative integers using the Fibonacci rule
Matrix form
A 2dimensional system of linear difference equations that describes the Fibonacci sequence is
or
The eigenvalues of the matrix A are and , and the elements of the eigenvectors of A, and , are in the ratios and .
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:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , 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:
Taking the determinant of both sides of this equation yields Cassini's identity
Additionally, since for any square matrix , the following identities can be derived:
Recognizing Fibonacci numbers
Occasionally, the question may arise whether a positive integer is a Fibonacci number. Since is the closest integer to , the most straightforward, bruteforce test is the identity
which is true if and only if is a Fibonacci number.
Alternatively, an elegant algebraic test states, that a positive integer is a Fibonacci number if (and only if) or is a perfect square.<ref>{{
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A slightly more sophisticated test uses the fact that the convergents of the continued fraction representation of are ratios of successive Fibonacci numbers, that is the inequality
(with coprime positive integers , ) is true if and only if and are successive Fibonacci numbers. From this one derives the criterion that is a Fibonacci number if and only if the closed interval
contains a positive integer.<ref>M. Möbius, Wie erkennt man eine Fibonacci Zahl?, Math. Semesterber. (1998) 45; 243–246</ref>
Identities
 F(n + 1) = F(n) + F(n − 1)
 F(0) + F(1) + F(2) + … + F(n) = F(n + 2) − 1
 F(1) + 2 F(2) + 3 F(3) + … + n F(n) = n F(n + 2) − F(n + 3) + 2
 F(0)² + F(1)² + F(2)² + … + F(n)² = F(n) F(n + 1)
These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using combinatorial arguments here. In particular, F(n) can be interpreted as the number of ways summing 1's and 2's to n − 1, with the convention that F(0) = 0, meaning no sum will add up to −1, and that F(1) = 1, meaning the empty sum will "add up" to 0. Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice.
Proof of the first identity
Without loss of generality, we may assume n ≥ 1. Then F(n + 1) counts the number of ways summing 1's and 2's to n.
When the first summand is 1, there are F(n) ways to complete the counting for n − 1; and when the first summand is 2, there are F(n − 1) ways to complete the counting for n − 2. Thus, in total, there are F(n) + F(n − 1) ways to complete the counting for n.
Proof of the second identity
We count the number of ways summing 1's and 2's to n + 1 such that at least one of the summands is 2.
As before, there are F(n + 2) ways summing 1's and 2's to n + 1 when n ≥ 0. Since there is only one sum of n + 1 that does not use any 2, namely 1 + … + 1 (n + 1 terms), we subtract 1 from F(n + 2).
Equivalently, we can consider the first occurrence of 2 as a summand. If, in a sum, the first summand is 2, then there are F(n) ways to the complete the counting for n − 1. If the second summand is 2 but the first is 1, then there are F(n − 1) ways to complete the counting for n − 2. Proceed in this fashion. Eventually we consider the (n + 1)th summand. If it is 2 but all of the previous n summands are 1's, then there are F(0) ways to complete the counting for 0. If a sum contains 2 as a summand, the first occurrence of such summand must take place in between the first and (n + 1)th position. Thus F(n) + F(n − 1) + … + F(0) gives the desired counting.
Proof of the third identity
This identity can be established in two stages. First, we count the number of ways summing 1s and 2s to −1, 0, …, or n + 1 such that at least one of the summands is 2.
By our second identity, there are F(n + 2) − 1 ways summing to n + 1; F(n + 1) − 1 ways summing to n; …; and, eventually, F(2) − 1 way summing to 1. As F(1) − 1 = F(0) = 0, we can add up all n + 1 sums and apply the second identity again to obtain
 [F(n + 2) − 1] + [F(n + 1) − 1] + … + [F(2) − 1]
 = [F(n + 2) − 1] + [F(n + 1) − 1] + … + [F(2) − 1] + [F(1) − 1] + F(0)
 = F(n + 2) + [F(n + 1) + … + F(1) + F(0)] − (n + 2)
 = F(n + 2) + F(n + 3) − (n + 2).
On the other hand, we observe from the second identity that there are
 F(0) + F(1) + … + F(n − 1) + F(n) ways summing to n + 1;
 F(0) + F(1) + … + F(n − 1) ways summing to n;
……
 F(0) way summing to −1.
Adding up all n + 1 sums, we see that there are
 (n + 1) F(0) + n F(1) + … + F(n) ways summing to −1, 0, …, or n + 1.
Since the two methods of counting refer to the same number, we have
 (n + 1) F(0) + n F(1) + … + F(n) = F(n + 2) + F(n + 3) − (n + 2)
Finally, we complete the proof by subtracting the above identity from n + 1 times the second identity.
Identity for doubling n
There is a very simple formula for doubling n :. <ref>Fibonacci Number  from Wolfram MathWorld</ref>
Another identity useful for calculating F_{n} for large values of n is
for all integers n and k. Dijkstra<ref>E. W. Dijkstra (1978). In honour of Fibonacci. Report EWD654.</ref> points out that doubling identities of this type can be used to calculate F_{n} 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) . Thus the actual performance depends mainly upon efficiency of the implemented long multiplication, and usually is or .)
Other identities
Other identities include relationships to the Lucas numbers, which have the same recursive properties but start with L_{0}=2 and L_{1}=1. These properties include F_{2n}=F_{n}L_{n}.
There are also scaling identities, which take you from F_{n} and F_{n+1} to a variety of things of the form F_{an+b}; for instance
by Cassini's identity.
These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number. Such relations exist in a very general sense for numbers defined by recurrence relations, see the section on multiplication formulae under Perrin numbers for details.
Power series
The generating function of the Fibonacci sequence is the power series
This series has a simple and interesting closedform solution for x < 1/
This solution can be proven by using the Fibonacci recurrence to expand each coefficient in the infinite sum defining :
Solving the equation for results in the closed form solution.
In particular, math puzzlebooks note the curious value , or more generally
for all integers .
Conversely,
Reciprocal sums
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every oddindexed reciprocal Fibonacci number as
and the sum of squared reciprocal Fibonacci numbers as
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
and there is a nice nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,
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
has been proved irrational by Richard AndréJeannin.
Primes and divisibility
 Main article: Fibonacci prime
A Fibonacci prime is a Fibonacci number that is prime (sequence A005478 in OEIS). The first few are:
 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many. They must all have a prime index, except F_{4} = 3.
Any three consecutive Fibonacci numbers, taken two at a time, are relatively prime: that is,
 gcd(F_{n},F_{n+1}) = gcd(F_{n},F_{n+2}) = 1.
More generally,
 gcd(F_{n}, F_{m}) = F_{gcd(n,m).}<ref>Paulo Ribenboim, My Numbers, My Friends, SpringerVerlag 2000</ref>
A proof of this striking fact is online at Harvey Mudd College's Fun Math site
Divisibility by prime numbers
If p is a prime number then<ref>Paulo Ribenboim (1996), The New Book of Prime Number Records, New York: Springer, ISBN 0387944575, p. 64</ref><ref>Franz Lemmermeyer (2000), Reciprocity Laws, New York: Springer, ISBN 3540669574, ex 2.252.28, pp. 7374</ref>
where is the Legendre symbol. if p ≡ ±2 (mod 5), if p ≡ ±1 (mod 5), and .
For example,
Divisibility by 11
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. The triangle sides a, b, c can be calculated directly:
These formulas satisfy for all n, but they only represent triangle sides when .
Magnitude of Fibonacci numbers
Since is asymptotic to , the number of digits in the base b representation of is asymptotic to .
In base 10, for every integer greater than 1 there are 4 or 5 Fibonacci numbers with that number of digits, in most cases 5.
Applications
The Fibonacci numbers are important in the runtime analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.
The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle and Lozanić's triangle (see "Binomial coefficient").
Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation.
Fibonacci numbers are used by some pseudorandom number generators.
Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers  by dividing the list so that the two parts have lengths in the approximate proportion φ. A tapedrive 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 onedimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers.<ref>{{#if:M. Avriel and D.J. Wilde
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In music, Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements. It is commonly thought that the first movement of Béla Bartók's Music for Strings, Percussion, and Celesta was structured using Fibonacci numbers.
Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio (denoted φ), the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.<ref>An Application of the Fibonacci Number Representation</ref><ref>A Practical Use of the Sequence</ref><ref>Zeckendorf representation</ref>
Fibonacci numbers in nature
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Przemyslaw Prusinkiewicz advanced the idea that real instances can be in part understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.<ref>{{
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A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.<ref> Template:Citation</ref> This has the form
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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>{{
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Popular culture
 Main article: Fibonacci numbers in popular culture
Because the Fibonacci sequence is easy for nonmathematicians to understand, there are many examples of the Fibonacci numbers being used in popular culture.
Generalizations
 Main article: Generalizations of Fibonacci numbers
The Fibonacci sequence has been generalized in many ways. These include:
 Extending to negative index n, satisfying F_{n} = F_{n−1} + F_{n−2} and, equivalently, F_{n} = (−1)^{n+1}F_{n}
 Generalising the index from positive integers to real numbers using a modification of Binet's formula. <ref>Template:MathWorld</ref>
 Starting with other integers. Lucas numbers have L_{1} = 1, L_{2} = 3, and L_{n} = L_{n−1} + L_{n−2}. Primefree sequences use the Fibonacci recursion with other starting points in order to generate sequences in which all numbers are composite.
 Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P_{n} = 2P_{n – 1} + P_{n – 2}.
 Not adding the immediately preceding numbers. The Padovan sequence and Perrin numbers have P(n) = P(n – 2) + P(n – 3).
 Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more.
 Adding other objects than integers, for example functions or strings  one essential example is Fibonacci polynomials.
Numbers properties
Periodicity mod n: Pisano periods
It is easily seen that if the members of the Fibonacci sequence are taken mod n, the resulting sequence must be periodic with period at most . The lengths of the periods for various n form the socalled Pisano periods (sequence A001175 in OEIS). Determining the Pisano periods in general is an open problem,{{#if:{{#if:Category:Articles with unsourced statements[[Category:Articles with unsourced statements {{#if:March 2008{{#if:fromsince}} March 2008}}]]}}}}{{#if:citation needed^{[citation needed]}}}{{#if:{{#if:March 2008{{#ifexist:Category:Articles with unsourced statements since March 2008}}}}}} although for any particular n it can be solved as an instance of cycle detection.
Pythagorean triples
Any four consecutive Fibonacci numbers F_{n}, F_{n+1}, F_{n+2} and F_{n+3} can be used to generate a Pythagorean triple:
Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then:
Example 2: let the Fibonacci numbers be 8, 13, 21 and 34. Then:
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>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>Template:Cite article</ref>
See also
 Logarithmic spiral
 b:Fibonacci number program at Wikibooks
 The Fibonacci Association
 Fibonacci Quarterly — an academic journal devoted to the study of Fibonacci numbers
 Negafibonacci numbers
 Lucas number
References
External links
 Peter Marcer, describing the discovery by jeanclaude Perez of Fibonacci numbers structuring proportions of TCAG nucleotides within DNA, (1992).
 Ron Knott, The Golden Section: Phi, (2005).
 Ron Knott, Representations of Integers using Fibonacci numbers, (2004).
 wallstreetcosmos.com, Fibonacci numbers and stock market analysis, (2008).
 Juanita Lofthouse Fibonacci numbers and Red Blood Cell Dynamics, .
 Bob Johnson, Fibonacci resources, (2004)
 Donald E. Simanek, Fibonacci FlimFlam, (undated, 2005 or earlier).
 Rachel Hall, Hemachandra's application to Sanskrit poetry, (undated; 2005 or earlier).
 Alex Vinokur, Computing Fibonacci numbers on a Turing Machine, (2003).
 (no author given), Fibonacci Numbers Information, (undated, 2005 or earlier).
 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 Fibonacci Association incorporated in 1963, focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas.
 Dawson Merrill's FibPhi link page.
 Fibonacci primes
 Periods of Fibonacci Sequences Mod m at MathPages
 The One Millionth Fibonacci Number
 The Ten Millionth Fibonacci Number
 An Expanded Fibonacci Series Generator
 Manolis Lourakis, Fibonaccian search in C
 Scientists find clues to the formation of Fibonacci spirals in nature
 Fibonacci Numbers at Convergence
 Online Fibonacci calculator