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Encyclopedia > Tetranacci number
A tiling with squares whose sides are successive Fibonacci numbers in length
A tiling with squares whose sides are successive Fibonacci numbers in length
A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above - see golden spiral.
A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above - see golden spiral.

In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: Image File history File links FibonacciBlocks. ... Image File history File links FibonacciBlocks. ... Image File history File links Fibonacci_spiral. ... Image File history File links Fibonacci_spiral. ... A golden spiral is a logarithmic spiral whose growth factor b is related to phi, the golden ratio. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

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 (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, … , are: The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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…

Sometimes this sequence is considered to start at F1 = 1, but it is more common to include F0 = 0.


The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India.[1][2] Leonardo of Pizza (1170s or 1180s – 1250), also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some the most talented mathematician of the Middle Ages. ...

Contents

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 Gopala in the 12th c. also revisits the problem in some detail. Sanskrit grammatical tradition (, one of the six Vedanga disciplines) begins in late Vedic India, and culminates in the AṣṭādhyāyÄ« of Pāṇini (ca. ... Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... Centuries: 6th century BC - 5th century BC - 4th century BC Decades: 500s BC 490s BC 480s BC 470s BC 460s BC - 450s BC - 440s BC 430s BC 420s BC 410s BC 400s BC Years: 455 BC 454 BC 453 BC 452 BC 451 BC - 450 BC - 449 BC 448 BC... Centuries: 3rd century BC - 2nd century BC - 1st century BC Decades: 250s BC 240s BC 230s BC 220s BC 210s BC - 200s BC - 190s BC 180s BC 170s BC 160s BC 150s BC Years: 205 BC 204 BC 203 BC 202 BC 201 BC - 200 BC - 199 BC 198 BC... In linguistics, prosody refers to intonation, rhythm, and vocal stress in speech. ... Here is a chronology of the main Indian mathematicians: BC Yajnavalkya, 1800 BC, the author of the altar mathematics of the Shatapatha Brahmana. ... Virahanka (विरहाङ्क) was an Indian prosodicist who is also known for his work on mathematics. ... JAIN is an activity within the Java Community Process, developing APIs for the creation of telephony (voice and data) services. ... Hemachandra SurÄ« ({{lang-sa|हेमचन्द्र सूरी) (1089–1172) was an Indian Jaina scholar, poet, and polymath who wrote on grammar, philosophy, prosody, and contemporary history. ... Events Åhus, Sweden gains city privileges City of Airdrie, Scotland founded King Sverker I of Sweden is deposed and succeeded by Eric IX of Sweden. ... Gopala was an Indian mathematician, who studied the Fibonacci numbers in 1135, more than half a century before Fibonacci popularized these numbers in Europe. ...


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: 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; 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. Mora (plural moras or morae) is a unit of sound used in phonology that determines syllable weight (which in turn determines stress) in some languages. ... Donald Knuth at a reception for the Open Content Alliance. ... Cover of books The Art of Computer Programming[1] is a comprehensive monograph written by Donald Knuth which covers many kinds of programming algorithms and their analysis. ... In computational complexity theory, the bin packing problem is a combinatorial NP-hard problem. ...


In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202)[3]. He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that: Leonardo of Pizza (1170s or 1180s – 1250), also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some the most talented mathematician of the Middle Ages. ... Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. ... // Events August 1 - Arthur of Brittany captured in Mirebeau, north of Poitiers Beginning of the Fourth Crusade. ...

  • 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).[4]


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 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). 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

The golden ratio.
The golden ratio.

The golden ratio varphi (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: The golden ratio (phi) represented as a line divided into two segments a and b, such that the entire line is to the longer a segment as the a segment is to the shorter b segment. ... The golden ratio (phi) represented as a line divided into two segments a and b, such that the entire line is to the longer a segment as the a segment is to the shorter b segment. ... The golden section is a line segment sectioned into two according to the golden ratio. ... Look up Φ, φ in Wiktionary, the free dictionary. ...

frac{x}{1}=frac{1}{x-1} or equivalently x^2-x-1=0,,

which is equal to varphi = frac{(1 + sqrt{5})}{2}approx 1.618,033,988,749,894,848,204,586,834,366,.


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: In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of well-known operations. ... Jacques Philippe Marie Binet was a catholic mathematician. ...

Fleft(nright) = {{varphi^n-(1-varphi)^n} over {sqrt 5}}, , where varphi is the golden ratio defined above.

Note the similarity of the Fibonacci recursion

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

to the defining equation of the golden ratio in the form

x^2-x-1=0,,

also known as the generating polynomial of the recursion.


Proof (by induction): Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...


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) = avarphi^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) &= avarphi^{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+2varphi}{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 < 1/2end{matrix} for all ngeq 0, , F(n), is the closest integer to varphi^n/sqrt 5, . For computational purposes, this is expressed using the floor function: The floor and fractional part functions In mathematics, the floor function of a real number x, denoted or floor(x), is the largest integer less than or equal to x (formally, ). For example, floor(2. ...

F(n)=bigglfloorfrac{varphi^n}{sqrt 5} + frac{1}{2}biggrfloor.

Limit of consecutive quotients

Johannes Kepler pointed out that the ratio of consecutive Fibonacci numbers converges to the golden ratio varphi as the limit. Johannes Kepler (December 27, 1571 – November 15, 1630) was a German Lutheran mathematician, astronomer and astrologer, and a key figure in the 17th century astronomical revolution. ...

lim_{ntoinfty}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_{ntoinfty}frac{F_{a,b}(n+1)}{F_{a,b}(n)} &= lim_{ntoinfty}frac{avarphi^{n+1}-b(1-varphi)^{n+1}}{avarphi^n-b(1-varphi)^n} &= lim_{ntoinfty}frac{avarphi-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_{ntoinfty}left(tfrac{1-varphi}{varphi}right)^n=0


Matrix form

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

{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,!). In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...


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: varphi,! = [1; 1, 1, 1, 1, …]. 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. In mathematics, a unimodular matrix is a square matrix with determinant +1 or -1. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...


The matrix representation gives the following closed expression for the Fibonacci numbers: In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of well-known operations. ...

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 Cassinis identity is a mathematical identity for the Fibonacci numbers. ...

F_{n+1}F_{n-1} - F_n^2 = (-1)^n.,

Additionally, since AnAm = Am + 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}.,

Recognising 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

Fbigg(bigglfloorlog_varphi(sqrt{5}z)+frac{1}{2}biggrfloorbigg)=z,

which is true if and only if z is a Fibonacci number. It has been suggested that this article or section be merged with Logical biconditional. ...


A slightly more sophisticated test uses the fact that the convergents of the continued fraction representation of varphi are ratios of successive Fibonacci numbers, that is the inequality A convergent is one of a sequence of rational values obtained by evaluating successive truncations of a continued fraction. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

bigg|varphi-frac{p}{q}bigg|<frac{1}{q^2}

(with coprime positive integers p, q) is true if and only if p and q are successive Fibonacci numbers. From this one derives the criterion that z is a Fibonacci number if and only if the intersection of the closed interval Coprime - Wikipedia /**/ @import /skins-1. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

bigg[varphi z-frac{1}{z},varphi z+frac{1}{z}bigg]

with the positive integers mathbb{N} is not empty.[5]


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. In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...


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. Yuri Matiyasevich born March 2, 1947 in Leningrad, is a Russian mathematician. ... In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ... Matiyasevichs theorem, proven in 1970 by Yuri Matiyasevich, implies that Hilberts tenth problem is unsolvable. ...


The Fibonacci numbers occur in the sums of diagonals in Pascal's triangle and Lozanić's triangle (see "Binomial coefficient"). 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first five rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ... Lozanićs triangle (sometimes called Losanitschs triangle) is a geometric arrangement of binomial coefficients in a manner very similar to that of Pascals triangle. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...


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. Zeckendorfs theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers. ...


Fibonacci numbers are used by some pseudorandom number generators. A pseudorandom number generator (PRNG) is an algorithm that generates a sequence of numbers, the elements of which are approximately independent of each other. ...


A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers.[6] The Fibonacci search technique is a method of searching a sorted array using a divide and conquer algorithm that narrows down possible locations with the aid of 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 formal elements. Examples include Béla Bartók's Music for Strings, Percussion, and Celesta. Allegory of Music on the Opéra Garnier Music is an art form that involves organized sounds and silence. ... Look up content in Wiktionary, the free dictionary. ... The term musical form is used in two related ways: a generic type of composition such as the symphony or concerto the structure of a particular piece, how its parts are put together to make the whole; this too can be generic, such as binary form or sonata form Musical... Béla Viktor János Bartók (March 25, 1881 – September 26, 1945) was a Hungarian composer, pianist and collector of Eastern European and Middle Eastern folk music. ... Music for Strings, Percussion and Celesta is a piece of classical music by Béla Bartók. ...


Since the conversion factor 1.609 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. Conversion of units refers to conversion factors between different units of measurement for the same quantity. ... A mile is a unit of length, usually used to measure distance, in a number of different systems, including Imperial units, United States customary units and Norwegian/Swedish mil. ... The golden section is a line segment sectioned into two according to the golden ratio. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... The Fibonacci code is a universal code which encodes positive integers into binary code words. ... In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to commonly used values&#8212;typically, the values being in the midst of a calculation at a given point in time. ... Golden ratio base refers to the use of the golden ratio, the irrational number ≈1. ...


Fibonacci numbers in nature

Sunflower head displaying florets in spirals of 34 and 55 around the outside
Sunflower head displaying florets in spirals of 34 and 55 around the outside

Fibonacci sequences appear in biological settings,[7] such as branching in trees, the curve of waves,[citation needed] the fruitlets of a pineapple, 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 groups, specifically as certain Lindenmayer grammars.[9] Image File history File links Helianthus_whorl. ... Image File history File links Helianthus_whorl. ... Binomial name Helianthus annuus L. The sunflower (Helianthus annuus) is an annual plant in the family Asteraceae, with a large flower head (inflorescence). ... This does not adequately cite its references or sources. ... A cone (in formal botanical usage: strobilus, plural strobili) is an organ on plants in the division Pinophyta (conifers) that contains the reproductive structures. ... A Polish mathematician who has advanced the idea that fibonacci numbers in nature can be in part understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmeyer grammars. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... See L-system for information on Lindenmayer systems. ...


Identities

  1. F(n + 1) = F(n) + F(n − 1)
  2. F(0) + F(1) + F(2) + … + F(n) = F(n + 2) − 1
  3. F(1) + 2 F(2) + 3 F(3) + … + n F(n) = n F(n + 2) − F(n + 3) + 2
  4. 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. A combinatorial proof is a method of proving a statement, usually a combinatorics identity, by counting some carefully chosen object in different ways to obtain different expressions in the statement (see also double counting). ...


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. Without loss of generality or simply WLOG is a frequently used expression in mathematics. ...


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.


Identity for doubling n

Another identity useful for calculating Fn for large values of n is

F_{2n+k} = F_k F_{n+1}^2 + 2 F_{k-1} F_{n+1} F_n + F_{k-2} F_n^2

for all integers n and k.


Other identities

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 F2n=FnLn The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. ...


Power series

The generating function of the Fibonacci sequence is the power series In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

s(x)=sum_{k=0}^{infty} F_k x^k.

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

s(x)=frac{x}{1-x-x^2}.

This solution can be proven by using the Fibonacci recurrence to expand each coefficient in the infinite sum defining s(x):

begin{align} s(x) &= sum_{k=0}^{infty} F_k x^k &= F_0 + F_1x + sum_{k=2}^{infty} left( F_{k-1} + F_{k-2} right) x^k &= x + sum_{k=2}^{infty} F_{k-1} x^k + sum_{k=2}^{infty} F_{k-2} x^k &= x + xsum_{k=0}^{infty} F_k x^k + x^2sum_{k=0}^{infty} F_k x^k &= x + x s(x) + x^2 s(x) end{align}

Solving the equation s(x) = x + xs(x) + x2s(x) for s(x) results in the closed form solution.


In particular, math puzzle-books note the curious value frac{s(frac{1}{10})}{10}=frac{1}{89}.


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 odd-indexed Fibonacci number as In mathematics, theta functions are special functions of several complex variables. ...

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_2^4left(0, frac{3-sqrt 5}{2}right) - vartheta_4^4left(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, The golden section is a line segment sectioned into two according to 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

C = sum_{k=1}^{infty} frac{1}{F_k} = 3.359885666243 dots

has been proved irrational by Richard André-Jeannin. In mathematics, an irrational number is any real number that is not a rational number, i. ...


Primes and divisibility

Main article: Fibonacci prime

A Fibonacci prime is a Fibonacci number that is prime A005478. The first few are: A Fibonacci prime is a Fibonacci number that is prime. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

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 are relatively prime: that is, In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and &#8722;1, or equivalently, if their greatest common divisor is 1. ...

gcd(Fn,Fn+1) = gcd(Fn,Fn+2) = 1.

More generally, In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

gcd(Fn, Fm) = Fgcd(n,m).[10]

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


Right Triangles

Starting with 5, every other Fibonacci number is the length of the hypotenuse of a right triangle with integral 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 Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ...


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.


Popular culture

Main article: Fibonacci numbers in popular culture

The Fibonacci sequence is easy to understand for non-mathematicians, contributing to the Fibonacci numbers being embraced by popular culture. They appear in novels (e.g. The Da Vinci Code which is also a film), films (e.g. Darren Aronofsky's π), and in various television shows (e.g. NUMB3RS and So Weird ). The Fibonacci numbers form a sequence of integers, mathematically defined by: F(0) = 0. ... This article is about the novel. ... Darren Aronofsky (born February 12, 1969 in Brooklyn, New York) is an American film director, screenwriter and film producer. ... Ï€ (or Pi) is a 1998 American psychological thriller directed by Darren Aronofsky. ... NUMB3RS (Numbers) is an American television show that follows FBI Special Agent Don Eppes (Rob Morrow) and his mathematical genius brother, Charlie Eppes (David Krumholtz), who develops formulae to predict the actions of various criminals. ... So Weird was a television series shot in Vancouver, British Columbia that aired on the Disney Channel from 1999 to 2001. ...


In addition to just mentioning Fibonacci numbers, they have also been applied to create architecture, visual art works, poetry, and music; sometimes through their relation to the golden ratio. The golden section is a line segment sectioned into two according to the golden ratio. ...


Generalizations

Main article: Generalizations of Fibonacci numbers

The Fibonacci sequence has been generalized in many ways. These include: In mathematics, the Fibonacci numbers form a sequence defined recursively by: F(0) = 0 F(1) = 1 F(n) = F(n-1) + F(n-2), for integer n > 1 That is, after two starting values, each number is the sum of the two preceding numbers. ...

  • Extending to negative index n, satisfying Fn = Fn-1 + Fn-2 and, equivalently, F-n = (-1)n+1Fn
  • Starting with other integers. Lucas numbers have L1 = 1, L2 = 3, and Ln = Ln-1 + Ln-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 Pn = 2Pn - 1 + Pn - 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.

The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. ... In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... In mathematics, the Pell numbers and companion Pell numbers (Pell-Lucas numbers) are both sequences of integers. ... The Padovan sequence is the sequence of integers P(n) defined by the initial values and the recurrence relation The first few values of P(n) are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 144, 151, 200. ... In mathematics, the Perrin numbers are defined by the recurrence relation P(0) = 3, P(1) = 0, P(2) = 2, and P(n) = P(n − 2) + P(n − 3) for n > 2. ...

See also

The golden section is a line segment sectioned into two according to the golden ratio. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ...

References

  1. ^ Parmanand Singh. Acharya Hemachandra and the (so called) Fibonacci Numbers. Math . Ed. Siwan , 20(1):28-30,1986.ISSN 0047-6269]
  2. ^ Parmanand Singh,"The So-called Fibonacci numbers in ancient and medieval India. Historia Mathematica v12 n3, 229-244,1985
  3. ^ Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. ISBN 0-387-95419-8.  Chapter II.12, pp. 404–405.
  4. ^ Knott, Ron. Fibonacci's Rabbits. University of Surrey School of Electronics and Physical Sciences.
  5. ^ M. Möbius, Wie erkennt man eine Fibonacci Zahl?, Math. Semesterber. (1998) 45; 243—246
  6. ^ M. Avriel and D.J. Wilde (1966). "Optimality of the Symmetric Fibonacci Search Technique". The Fibonacci Quarterly (3): 265—269. 
  7. ^ S. Douady and Y. Couder (1996). "Phyllotaxis as a Dynamical Self Organizing Process". Journal of Theoretical Biology (178): 255–274. 
  8. ^ A. Brousseau (1969). "Fibonacci Statistics in Conifers". The Fibonacci Quarterly (7): 525—532. 
  9. ^ Prusinkiewicz, Przemyslaw; James Hanan (1989). Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics). Springer-Verlag. ISBN 0-387-97092-4. 
  10. ^ Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000

The University of Surrey (UniS) received its charter on September 9, 1966, and was at that time situated near Battersea Park in south-west London. ... Springer Science+Business Media or Springer (IPA: ) is a worldwide publishing company based in Germany which focuses on academic journals and books in the fields of science, technology and medicine. ... Paulo Ribenboim is a Canadian mathematician who speciliazes in number theory. ...

External links


  Results from FactBites:
 
What's Special About This Number? (7272 words)
is the number of planar partitions of 10.
is the number of planar partitions of 11.
is the number of planar partitions of 12.
  More results at FactBites »


 

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