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In mathematics, the Fibonacci numbers form a sequence defined recursively by:

In other words, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS) 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,

The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India.^[1]

Contents 1 Origins 2 The bee ancestry code 3 Relation to the golden ratio 3.1 Closed form expression 3.2 Limit of consecutive quotients 4 Matrix form 5 Applications 6 Fibonacci numbers in nature 7 Identities 7.1 Common factors 8 Power series 9 Reciprocal sums 10 Generalizations 10.1 Vector space 10.2 Similar integer sequences 10.2.1 Lucas numbers 10.2.2 Tribonacci numbers 10.2.3 Tetranacci numbers 10.2.4 Other -anacci numbers 10.3 Other generalizations 11 Fibonacci primes 12 Fibonacci strings 13 Popular culture 13.1 Architecture 13.2 Cinema 13.3 Literature 13.4 Music 13.5 Television 13.6 Visual Arts 14 See also 15 References 16 External links

Origins

The "Fibonacci" numbers first appear, under the name maatraameru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chhandah-shāstra, the Art of Prosody, 450 or 200 BC). The Indian mathematician Virahanka gave explicit rules for the Fibonacci sequence in the 8th century. The Indian Jain philosopher Hemachandra (c.1150) (and also Gopala) revisited the problem in some detail. Sanskrit vowel sounds can be long (L) or short (S), and Hemachandra wished to compute how many cadences of a given overall length can be composed of these. 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)

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; thus Hemachandra 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 his The Art of Computer Programming as equivalent to the problem of bin packing items of length 1 and 2.

In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci (1202). 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

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).

The bee ancestry code

Fibonacci is also stated as having described the sequence "encoded in the ancestry of a male bee." This turns out to be the Fibonacci sequence. One can derive this from the following sequence:

• If an egg is laid by a single female, it hatches a male.
• If, however, the egg is 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 this 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). If one continues this sequence, it gives a perfectly accurate depiction of the Fibonacci sequence.

However, this statement is mostly theoretical. 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

The golden ratio (phi), 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 mathematically, normalising the larger part to unit length, it is the positive solution of the equation:

or equivalently

which is equal to .

Closed form expression

Like every sequence defined by linear recursion, the Fibonacci numbers have a closed-form solution. It has become known as Binet's formula:

, where is the golden ratio defined above.

Note the similarity between the Fibonacci recursion:

to the defining equation of the golden ratio in the form

also known as the generating polynomial of the recursion.

Proof (by induction):

Any root of the equation above satifies and multiplying by shows:

Note that by definition is a root of the equation and that the other root is . Therefore:

	= and
	=

Now consider the functions:

defined for any real

All these functions satisfy the Fibonacci recursion

Selecting and gives the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore:

and

establishing the base cases of the induction, proving that

for all n.

Note that for any two starting values, a combination a,b can be found such that the function is the exact closed formula for the series.

In fact, since for all , is the closest integer to . For computational purposes, this is expressed using the floor function,

Limit of consecutive quotients

As was pointed out by Johannes Kepler, the ratio of consecutive Fibonacci numbers, that is:

converges to the golden ratio (phi)

Actually, this limit behaviour does not depend on the particular starting values (except 0, 0).

Proof:

It follows from the explicit formula that for any real :

	,

because, as is easily shown, and thus

Matrix form

A 2-dimensional 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 .

Note that 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 mean:  = [1; 1, 1, 1, 1, …]. 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 the identity

Additionally, since AⁿA^m = A^{m + n} for any square matrix A, the following identities can be derived:

Applications

The Fibonacci numbers are important in the run-time 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 a formula about the diagonals of Pascal'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.

A one-dimensional optimization method, called the Fibonacci search technique uses Fibonacci numbers ^[2].

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. Examples include Béla Bartók's Music for Strings, Percussion, and Celesta. In addition, the syllables of the lyrics of parts of the Tool song Lateralus follow the Fibonacci sequence in each line, for instance "Black/Then/White are/All I see/In my infancy/Red and yellow then came to be".

Since the conversion factor 1.609 for miles to kilometers is close to the golden mean 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 base φ being shifted. To go from kilometers to miles shift the register down the Fibonacci sequence instead.

Fibonacci numbers in nature

Fibonacci sequences have been noted to appear in biological settings,^[3] such as branching in trees and the arrangement of a pine cone^[4]. Przemyslaw Prusinkiewicz has advanced the idea that these can be in part understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.^[5]

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

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 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.

Common factors

Any two consecutive Fibonacci numbers are relatively prime. Suppose that F_n and F_n+1 have a common factor g. Then F_n−1 = F_n+1 – F_n must also be a multiple of g; and by induction the same must be true of all lower Fibonacci numbers. But F₁ = 1, so g = 1.

Other identities include relationships to the Lucas numbers, which have the same recursive properties but start with L₀=2 and L₁=1. These properties include F_2n=F_nL_n

Power series

The Fibonacci power series

has a simple and interesting closed-form solution for x < 1/φ:

This function is therefore the generating function of the Fibonacci sequence. It can be proven as follows:

Substituting F_k = F_{k − 1} + F_{k − 2}:

Therefore,

In particular, math puzzle-books note the curious value . The sum is easily proved by noting that

and then explicitly evaluating the sum.

Reciprocal sums

Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions.

The reciprocal Fibonacci constant

A079586

has been proved irrational by Richard André-Jeannin, but no closed form is known.

Generalizations

Vector space

The term Fibonacci sequence is also applied more generally to any function g where g(n + 2) = g(n) + g(n + 1). These functions are precisely those of the form g(n) = aF(n) + bF(n + 1) for some numbers a and b, so the Fibonacci sequences form a vector space with the functions F(n) and F(n + 1) as a basis.

Similar integer sequences

Lucas numbers

In particular, the Fibonacci sequence L with L(1) = 1 and L(2) = 3 is referred to as the Lucas numbers, after Edouard Lucas. This sequence was described by Leonhard Euler in 1748, in the Introductio in Analysin Infinitorum. The significance in the Lucas numbers L(n) lies in the fact that raising the golden ratio to the nth power yields

Lucas numbers are related to Fibonacci numbers by the relation

A generalization of the Fibonacci sequence are the Lucas sequences. One kind can be defined thus:

U(0) = 0

U(1) = 1

U(n + 2) = PU(n + 1) − QU(n)

where the normal Fibonacci sequence is the special case of P = 1 and Q = −1. Another kind of Lucas sequence begins with V(0) = 2, V(1) = P. Such sequences have applications in number theory and primality proving.

The Padovan sequence is generated by the recurrence P(n) = P(n − 2) + P(n − 3).

Tribonacci numbers

The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are A000073:

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, …

The tribonacci constant is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x³ − x² − x − 1, approximately 1.83929, and also satisfies the equation x + x⁻³ = 2. It is important in the study of the snub cube.

The tribonacci numbers are also given by

where the outer brackets denote the nearest integer function and

(Simon Plouffe, 1993).[1]

Tetranacci numbers

The tetranacci numbers start with four predetermined terms, each term afterwards being the sum of the preceding four terms. The first few tetranacci numbers are A000078:

0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, …

The tetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is a root of the polynomial x⁴ − x³ − x² − x − 1, approximately 1.92756, and also satisfies the equation x + x⁻⁴ = 2.

Other -anacci numbers

Pentanacci, hexanacci and heptanacci numbers have been computed, but they have not been of much interest to researchers.^{[citation needed]}

Other generalizations

The Fibonacci polynomials are another generalization of Fibonacci numbers.

A random Fibonacci sequence can be defined by tossing a coin for each position n of the sequence and taking F(n)=F(n−1)+F(n−2) if it lands heads and F(n)=F(n−1)−F(n−2) if it lands tails. Work by Furstenburg and Kesten guarantees that this sequence almost surely grows exponentially at a constant rate: the constant is independent of the coin tosses and was computed in 1999 by Divakar Viswanath. It is now known as Viswanath's constant.

A repfigit or Keith number is an integer, that when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4,7,11,18,29,47) reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the number, a tetranacci number if the number has four digits, etc. The first few repfigits are A007629:

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, …

Since the set of sequences satisfying the relation S(n) = S(n−1) + S(n−2) is closed under termwise addition and under termwise multiplication by a constant, it can be viewed as a vector space. Any such sequence is uniquely determined by a choice of two elements, so the vector space is two-dimensional. If we abbreviate such a sequence as (S(0), S(1)), the Fibonacci sequence F(n) = (0, 1) and the shifted Fibonacci sequence F(n−1) = (1, 0) are seen to form a canonical basis for this space, yielding the identity:

S(n) = S(0)F(n−1) + S(1)F(n)

for all such sequences S. For example, if S is the Lucas sequence 1, 3, 4, 7, 11…, then we obtain L(n) = F(n−1) + 3F(n).

Fibonacci primes

The first few Fibonacci numbers that are also prime numbers are A005478: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …. It is not known if there are infinitely many Fibonacci primes.

Fibonacci strings

In analogy to its numerical counterpart, a Fibonacci string is defined by:

,

where + denotes the concatenation of two strings. The sequence of Fibonacci strings starts:

b, a, ab, aba, abaab, abaababa, abaababaabaab, …

The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number.

Fibonacci strings appear as inputs for the worst case in some computer algorithms.

Popular culture

Architecture

• The sequence has been used in the design of a building, the Core, at the Eden Project, near St Austell, Cornwall, England.

Cinema

• Referenced in the film Dopo Mezzanotte (After Midnight) where the sequence appears as neon numbers on the dome of the Mole Antonelliana in Turin, Italy and is also used to select numbers in a lottery, ultimately winning it.
• Along with the concepts of the golden rectangle and golden spiral, the fibonacci sequence is used in Darren Aronofsky's independent film π (1998)
• Referenced in the film of The Phantom Tollbooth.
• It was also used as a key plot point in an episode of the Disney original television series So Weird.
• Used in Steven Spielberg's miniseries Taken.
• In The Da Vinci Code (film) an encrypted message was firstly thought to represent the Fibonacci numbers.

Literature

• The fibonacci sequence plays a small part in the bestselling novel and film The Da Vinci Code
• Fibs (poems of a specific form as per the fibonacci sequence) have been popularized by Gregory K. Pincus on his blog, Gottabook.
• A part of the fibonacci sequence is used as a code in Matthew Reilly's novel Ice Station.
• The sequence is used in the novel The Wright 3 by Blue Balliett.
• In Phillip K. Dick's novel VALIS, the Fibonnacci sequence (as well as the Fibonnacci constant) are used as identification signs by an organization called the "Friends of God".

Music

• MC Paul Barman structured the rhymes in his song "Enter Pan-Man" according to the fibonacci sequence. [2]
• Dr. Steel released a song titled "Fibonacci Sequence" in 2005.
• BT (Brian Transeau) released a dance track in 2000, entitled the "Fibonacci Sequence," which features a sample of a reading of the sequence.
• Tool's song "Lateralus" from the album of the same name features the fibonacci sequence symbolically in the verses of the song. The syllables in the first verse count 1, 1, 2, 3, 5, 8, 5, 3, 13, 8, 5, 3. Similarly, on Tool's 10,000 Days album there has already been speculation to more fibonacci references embedded within the album.
• The ratios of justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of the Fibonacci sequence
• Ernő Lendvai (1971) analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale. In Bartok's Music for Strings, Percussion and Celeste the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.^[6]
• French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.
• The Fibonacci numbers are also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which the sequence of keys is marked out by the intervals 34, 21, 13 and 8.^[6]

Television

• The fibonacci sequence is a key plot point in the television show MathNet's episode "The Case of the Willing Parrot."
• The fibonacci sequence is also referenced to in Numb3rs, the television series. Many times the cast reference note the relationship the sequence has with nature to further emphasise the wonders of mathematics.

Visual Arts

• In a FoxTrot comic, Jason and Marcus are playing football. Jason yells, "Hut 0! Hut 1! Hut 1! Hut 2!" all the way until "Hut 13!" in the fibonacci sequence. Marcus yells, "Is it the fibonacci sequence?" Jason says, "Correct! Touchdown, Marcus!"
• Marilyn Manson is another artist who has employed the Fibonacci sequence. He uses the sequence overtly in a watercolor painting entitled Fibonacci during his Holy Wood era, which it should be noted, uses bees as focal points. More discreetly, Manson used the sequence in the interior album art of Antichrist Superstar in his depiction of "The Vitruvian Man", in the vein of Leonardo DaVinci's work which was also based on the sequence. There is also speculation that some of the beats in the songs on the album Holy Wood (In the Shadow of the Valley of Death) are based on the Fibonacci sequence as well.
• Mario Merz frequently uses the fibonacci sequence in his art work

See also

• Fibonacci number program
• Anti-Fibonacci numbers
• Golden ratio (or golden section)
• Logarithmic spiral
• Plastic number
• Padovan number
• Perrin number

References

External links

• The Fibonacci Quarterly — an academic journal devoted to the study of Fibonacci numbers
• Alexey Stakhov, Museum of Harmony and Golden Section, (undated, 2005 or earlier).
• Subhash Kak, The Golden Mean and the Physics of Aesthetics, Archive of Physics, (2004).
• Ron Knott, The Golden Section: Phi, (2005).
• Ron Knott, Representations of Integers using Fibonacci numbers, (2004).
• Bob Johnson, Fibonacci resources, (2004)
• Donald E. Simanek, Fibonacci Flim-Flam, (undated, 2005 or earlier).
• Rachel Hall, Hemachandra's application to Sanskrit poetry, (undated; 2005 or earlier).
• Schweizer, David. The Fibonacci Numbers, (1996).
• Alex Vinokur, Computing Fibonacci numbers on a Turing Machine, (2003).
• (no author given), Fibonacci Numbers Information, (undated, 2005 or earlier).
• Fibonacci resources site
• 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 Fib-Phi link page.
• Fibonacci primes
• The One Millionth Fibonacci Number
• Math for Poets and Drummers - Rachael Hall surveys rhythm and Fibonacci numbers and also the Hemachandra connection. Saint Joseph's University, 2005.
• Source code for calculating the Fibonacci sequence in many programming languages.
• Actionscript 2.0 Fibonacci sequence generator class
• Fibonacci Calculator and Pre-caculated Fibonacci levels - examples of how Fibonacci's work is used every day in the futures markets.