Fibonacci number

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In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,[1]

A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21.
and
for n > 1.

The sequence starts:[2]

Under some older definitions, the value is omitted, so that the sequence starts with and the recurrence is valid for n > 2.[3][4] In his original definition, Fibonacci started the sequence with [5]

The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; (see preceding image)

Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.

Fibonacci numbers are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics,[6] although the sequence had been described earlier in Indian mathematics,[7][8][9] as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.

They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, and the arrangement of a pine cone's bracts.

Fibonacci numbers are also closely related to Lucas numbers , in that the Fibonacci and Lucas numbers form a complementary pair of Lucas sequences: and .

HistoryEdit

 
Thirteen (F7) ways of arranging long (shown by the red tiles) and short syllables (shown by the grey squares) in a cadence of length six. Five (F5) end with a long syllable and eight (F6) end with a short syllable.

The Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1986.[8][10][11] In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1.[9]

Knowledge of the Fibonacci sequence was expressed as early as Pingala (c. 450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for m beats (Fm+1) is obtained by adding one [S] to the Fm cases and one [L] to the Fm−1 cases.[12]Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).[13][7] However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):[11]

Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].[a]

Hemachandra (c. 1150) is credited with knowledge of the sequence as well,[7] writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."[15][16]

 
A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.
 
The number of rabbit pairs form the Fibonacci sequence

Outside India, the Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci[6][17] where it is used to calculate the growth of rabbit populations.[18][19] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?

  • At the end of the first month, they mate, but there is still only 1 pair.
  • At the end of the second month they produce a new pair, so there are 2 pairs in the field.
  • At the end of the third month, the original pair produce a second pair, but the second pair only mate without breeding, so there are 3 pairs in all.
  • At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.

At the end of the nth month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month n – 2) plus the number of pairs alive last month (month n – 1). The number in the nth month is the nth Fibonacci number.[20]

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.[21]

Sequence propertiesEdit

The first 21 Fibonacci numbers Fn are:[2]

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

The sequence can also be extended to negative index n using the re-arranged recurrence relation

 

which yields the sequence of "negafibonacci" numbers[22] satisfying

 

Thus the bidirectional sequence is

F−8 F−7 F−6 F−5 F−4 F−3 F−2 F−1 F0 F1 F2 F3 F4 F5 F6 F7 F8
−21 13 −8 5 −3 2 −1 1 0 1 1 2 3 5 8 13 21

Relation to the golden ratioEdit

Closed-form expressionEdit

Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression. It has become known as Binet's formula, named after French mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli:[23]

 
where
 
is the golden ratio (OEISA001622), and[24]
 

Since  , this formula can also be written as

 
To see this,[25] note that φ and ψ are both solutions of the equations
 
so the powers of φ and ψ satisfy the Fibonacci recursion. In other words,
 
and
 

It follows that for any values a and b, the sequence defined by

 
satisfies the same recurrence.
 

If a and b are chosen so that U0 = 0 and U1 = 1 then the resulting sequence Un must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:

 
which has solution
 
producing the required formula.

Taking the starting values U0 and U1 to be arbitrary constants, a more general solution is:

 
where
 
 

Computation by roundingEdit

Since

 

for all n ≥ 0, the number Fn is the closest integer to  . Therefore, it can be found by rounding, using the nearest integer function:

 

In fact, the rounding error is very small, being less than 0.1 for n ≥ 4, and less than 0.01 for n ≥ 8.

Fibonacci numbers can also be computed by truncation, in terms of the floor function:

 

As the floor function is monotonic, the latter formula can be inverted for finding the index n(F) of the largest Fibonacci number that is not greater than a real number F > 1:

 
where  

Limit of consecutive quotientsEdit

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 these ratios approach the golden ratio  [26][27]

 

This convergence holds regardless of the starting values, excluding 0 and 0, or any pair in the conjugate golden ratio,  [clarification needed] This can be verified using Binet's formula. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.

 
Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous

Decomposition of powersEdit

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:

 
This equation can be proved by induction on n.

This expression is also true for n < 1 if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule  

Matrix formEdit

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

 
alternatively denoted
 

which yields  . The eigenvalues of the matrix A are   and   corresponding to the respective eigenvectors

 
and
 
As the initial value is
 
it follows that the nth term is
 
From this, the nth element in the Fibonacci series may be read off directly as a closed-form expression:
 

Equivalently, the same computation may performed by diagonalization of A through use of its eigendecomposition:

 
where   and   The closed-form expression for the nth element in the Fibonacci series is therefore given by

 

which again yields

 

The matrix A 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 φ, 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-form expression for the Fibonacci numbers:

 

Taking the determinant of both sides of this equation yields Cassini's identity,

 

Moreover, since An Am = An+m for any square matrix A, the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing n into n + 1),

 

In particular, with m = n,

 

These last two identities provide a way to compute Fibonacci numbers recursively in O(log(n)) arithmetic operations and in time O(M(n) log(n)), where M(n) is the time for the multiplication of two numbers of n digits. This matches the time for computing the nth Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization).[28]

IdentificationEdit

The question may arise whether a positive integer x is a Fibonacci number. This is true if and only if at least one of   or   is a perfect square.[29] This is because Binet's formula above can be rearranged to give

 

which allows one to find the position in the sequence of a given Fibonacci number.

This formula must return an integer for all n, so the radical expression must be an integer (otherwise the logarithm does not even return a rational number).

Combinatorial identitiesEdit

Combinatorial proofsEdit

Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that Fn can be interpreted as the number of sequences of 1s and 2s that sum to n − 1. This can be taken as the definition of Fn, with the convention that F0 = 0, meaning no sum adds up to −1, and that F1 = 1, meaning the empty sum "adds up" to 0. Here, the order of the summand matters. For example, 1 + 2 and 2 + 1 are considered two different sums.

For example, the recurrence relation

 

or in words, the nth Fibonacci number is the sum of the previous two Fibonacci numbers, may be shown by dividing the Fn sums of 1s and 2s that add to n − 1 into two non-overlapping groups. One group contains those sums whose first term is 1 and the other those sums whose first term is 2. In the first group the remaining terms add to n − 2, so it has Fn-1 sums, and in the second group the remaining terms add to n − 3, so there are Fn−2 sums. So there are a total of Fn−1 + Fn−2 sums altogether, showing this is equal to Fn.

Similarly, it may be shown that the sum of the first Fibonacci numbers up to the nth is equal to the (n + 2)-nd Fibonacci number minus 1.[30] In symbols:

 

This is done by dividing the sums adding to n + 1 in a different way, this time by the location of the first 2. Specifically, the first group consists of those sums that start with 2, the second group those that start 1 + 2, the third 1 + 1 + 2, and so on, until the last group, which consists of the single sum where only 1's are used. The number of sums in the first group is F(n), F(n − 1) in the second group, and so on, with 1 sum in the last group. So the total number of sums is F(n) + F(n − 1) + ... + F(1) + 1 and therefore this quantity is equal to F(n + 2).

A similar argument, grouping the sums by the position of the first 1 rather than the first 2, gives two more identities:

 
and
 
In words, the sum of the first Fibonacci numbers with odd index up to F2n−1 is the (2n)th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to F2n is the (2n + 1)th Fibonacci number minus 1.[31]

A different trick may be used to prove

 
or in words, the sum of the squares of the first Fibonacci numbers up to Fn is the product of the nth and (n + 1)th Fibonacci numbers. In this case Fibonacci rectangle of size Fn by F(n + 1) can be decomposed into squares of size Fn, Fn−1, and so on to F1 = 1, from which the identity follows by comparing areas (see the first image at the top of this article).

Symbolic methodEdit

The sequence   is also considered using the symbolic method.[32] More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is  . Indeed, as stated above, the  -th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of   using terms 1 and 2.

It follows that the ordinary generating function of the Fibonacci sequence, i.e.  , is the complex function  .

Induction proofsEdit

Fibonacci identities often can be easily proved used mathematical induction.

For example, reconsider

 

Adding   to both sides gives

 

and so we have the formula for  

 

Similarly, add   to both sides of

 

to give

 
 

Binet formula proofsEdit

The Binet formula is

 

This can be used to prove Fibonacci identities.

For example, to prove that  

note that the left hand side multiplied by   becomes

 
 
 
 

As   and  , we have

 
 

as required.

Other identitiesEdit

Numerous other identities can be derived using various methods. Here are some of them:[33]

Cassini's and Catalan's identitiesEdit

Cassini's identity states that

 
Catalan's identity is a generalization:
 

d'Ocagne's identityEdit

 
 
where Ln is the n'th Lucas number. The last is an identity for doubling n; other identities of this type are
 
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.

More generally,[33]

 

or alternatively

 

Putting k = 2 in this formula, one gets again the formulas of the end of above section Matrix form.

Generating functionEdit

The generating function of the Fibonacci sequence is the power series

 

This series is convergent for   and its sum has a simple closed-form:[34]

 

This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum:

 

Solving the equation

 

for s(x) results in the closed form.

  gives the generating function for the negafibonacci numbers, and   satisfies the functional equation

 .

The partial fraction decomposition is given by

 

where   is the golden ratio and   is its conjugate.

Reciprocal sumsEdit

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

 

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 nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,

 

No closed formula for the reciprocal Fibonacci constant

 

is known, but the number has been proved irrational by Richard André-Jeannin.[35]

The Millin series gives the identity[36]

 
which follows from the closed form for its partial sums as N tends to infinity:
 

Primes and divisibilityEdit

Divisibility propertiesEdit

Every third number of the sequence is even and more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property[37][38]

 

Any three consecutive Fibonacci numbers are pairwise coprime, which means that, for every n,

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

Every prime number p divides a Fibonacci number that can be determined by the value of p modulo 5. If p is congruent to 1 or 4 (mod 5), then p divides Fp − 1, and if p is congruent to 2 or 3 (mod 5), then, p divides Fp + 1. The remaining case is that p = 5, and in this case p divides Fp.

 

These cases can be combined into a single, non-piecewise formula, using the Legendre symbol:[39]

 

Primality testingEdit

The above formula can be used as a primality test in the sense that if

 
where the Legendre symbol has been replaced by the Jacobi symbol, then this is evidence that n is a prime, and if it fails to hold, then n is definitely not a prime. If n is composite and satisfies the formula, then n is a Fibonacci pseudoprime. When m is large – say a 500-bit number – then we can calculate Fm (mod n) efficiently using the matrix form. Thus

 
Here the matrix power Am is calculated using modular exponentiation, which can be adapted to matrices.[40]

Fibonacci primesEdit

A Fibonacci prime is a Fibonacci number that is prime. The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... OEISA005478.

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.[41]

Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number.[42]

The only nontrivial square Fibonacci number is 144.[43] Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.[44] In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.[45]

1, 3, 21, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.[46]

No Fibonacci number can be a perfect number.[47] More generally, no Fibonacci number other than 1 can be multiply perfect,[48] and no ratio of two Fibonacci numbers can be perfect.[49]

Prime divisorsEdit

With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).[50] As a result, 8 and 144 (F6 and F12) are the only Fibonacci numbers that are the product of other Fibonacci numbers OEISA235383.

The divisibility of Fibonacci numbers by a prime p is related to the Legendre symbol   which is evaluated as follows:

 

If p is a prime number then

 
[51][52]

For example,

 

It is not known whether there exists a prime p such that

 

Such primes (if there are any) would be called Wall–Sun–Sun primes.

Also, if p ≠ 5 is an odd prime number then:[53]

 

Example 1. p = 7, in this case p ≡ 3 (mod 4) and we have:

 
 
 

Example 2. p = 11, in this case p ≡ 3 (mod 4) and we have:

 
 
 

Example 3. p = 13, in this case p ≡ 1 (mod 4) and we have:

 
 
 

Example 4. p = 29, in this case p ≡ 1 (mod 4) and we have:

 
 
 

For odd n, all odd prime divisors of Fn are congruent to 1 modulo 4, implying that all odd divisors of Fn (as the products of odd prime divisors) are congruent to 1 modulo 4.[54]

For example,

 

All known factors of Fibonacci numbers F(i) for all i < 50000 are collected at the relevant repositories.[55][56]

Periodicity modulo nEdit

If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n.[57] The lengths of the periods for various n form the so-called Pisano periods OEISA001175. Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.

MagnitudeEdit

Since Fn is asymptotic to  , the number of digits in Fn is asymptotic to  . As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits.

More generally, in the base b representation, the number of digits in Fn is asymptotic to  

GeneralizationsEdit

The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients.

Some specific examples that are close, in some sense, from Fibonacci sequence include:

  • Generalizing the index to negative integers to produce the negafibonacci numbers.
  • Generalizing the index to real numbers using a modification of Binet's formula.[33]
  • 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 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. If the coefficient of the preceding value is assigned a variable value x, the result is the sequence of Fibonacci polynomials.
  • 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. The resulting sequences are known as n-Step Fibonacci numbers.[58]

ApplicationsEdit

 
The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of Pascal's triangle.

The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):[59]

MathematicsEdit

 
 
Use of the Fibonacci sequence to count {1, 2}-restricted compositions

These numbers also give the solution to certain enumerative problems,[60] the most common of which is that of counting the number of ways of writing a given number n as an ordered sum of 1s and 2s (called compositions); there are Fn+1 ways to do this. For example, there are F5+1 = F6 = 8 ways one can climb a staircase of 5 steps, taking one or two steps at a time:

5 = 1+1+1+1+1 = 2+1+1+1 = 1+2+1+1 = 1+1+2+1 = 2+2+1
= 1+1+1+2 = 2+1+2 = 1+2+2

The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied recursively until a single step, of which there is only one way to climb.

To see how the formula is used, we can arrange the sums by the number of terms present:

5 = 1+1+1+1+1
= 2+1+1+1 = 1+2+1+1 = 1+1+2+1 = 1+1+1+2
= 2+2+1 = 2+1+2 = 1+2+2

which is  , where we are choosing the positions of k twos from n terms.

The Fibonacci numbers can be found in different ways among the set of binary strings, or equivalently, among the subsets of a given set.

  • The number of binary strings of length n without consecutive 1s is the Fibonacci number Fn+2. For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Equivalently, Fn+2 is the number of subsets S of {1, ..., n} without consecutive integers, that is, those S for which {i, i + 1} ⊈ S for every i. A bijection with the sums to n+1 is to replace 1 with 0 and 2 with 10, and drop the last zero.
  • The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number Fn+1. For example, out of the 16 binary strings of length 4, there are F5 = 5 without an odd number of consecutive 1s – they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets S of {1, ..., n} without an odd number of consecutive integers is Fn+1. A bijection with the sums to n is to replace 1 with 0 and 2 with 11.
  • The number of binary strings of length n without an even number of consecutive 0s or 1s is 2Fn. For example, out of the 16 binary strings of length 4, there are 2F4 = 6 without an even number of consecutive 0s or 1s – they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
  • Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his solving Hilbert's tenth problem.[61]
  • The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
  • Moreover, 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. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.
  • 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, obtained from the formula
     
    The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... The middle side of each of these triangles is the sum of the three sides of the preceding triangle.[62]
  • The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.
  • Fibonacci numbers appear in the ring lemma, used to prove connections between the circle packing theorem and conformal maps.[63]

Computer scienceEdit

 
Fibonacci tree of height 6. Balance factors green; heights red.
The keys in the left spine are Fibonacci numbers.
  • The Fibonacci numbers are important in computational 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.[64]
  • 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 tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming.
  • A Fibonacci tree is a binary tree whose child trees (recursively) differ in height by exactly 1. So it is an AVL tree, and one with the fewest nodes for a given height — the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.[65]
  • 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.[66]
  • The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law.[67][68]
  • They are also used in planning poker, which is a step in estimating in software development projects that use the Scrum methodology.

NatureEdit

 
Yellow chamomile head showing the arrangement in 21 (blue) and 13 (aqua) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.

Fibonacci sequences appear in biological settings,[69] such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[70] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone,[71] and the family tree of honeybees.[72][73] Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.[74] Field daisies most often have petals in counts of Fibonacci numbers.[75] In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.[76]

Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.[77]

 
Illustration of Vogel's model for n = 1 ... 500

A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel [de] in 1979.[78] This has the form

 

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. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,[79] typically counted by the outermost range of radii.[80]

Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules:

  • If an egg is laid by an unmated female, it hatches a male or drone bee.
  • If, however, an egg was fertilized by a male, it hatches a female.

Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, Fn, is the number of female ancestors, which is Fn−1, plus the number of male ancestors, which is Fn−2.[81] This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

 
The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships".[82])

It has been noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence.[82] A male individual has an X chromosome, which he received from his mother, and a Y chromosome, which he received from his father. The male counts as the "origin" of his own X chromosome ( ), and at his parents' generation, his X chromosome came from a single parent ( ). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome ( ). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome ( ). Five great-great-grandparents contributed to the male descendant's X chromosome ( ), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a population founder appears on all lines of the genealogy.)

The pathways of tubulins on intracellular microtubules arrange in patterns of 3, 5, 8 and 13.[83]

OtherEdit

  • In optics, when a beam of light shines at an angle through two stacked transparent plates of different materials of different refractive indexes, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have k reflections, for k > 1, is the  th Fibonacci number. (However, when k = 1, there are three reflection paths, not two, one for each of the three surfaces.)[84]
  • Fibonacci retracement levels are widely used in technical analysis for financial market trading.
  • Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio, 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.[85]
  • Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics.[86] In particular, it is shown how a generalised Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.
  • Mario Merz included the Fibonacci sequence in some of his artworks beginning in 1970.[87]
  • Joseph Schillinger (1895–1943) developed a system of composition which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.[88] See also Golden ratio § Music.

See alsoEdit

ReferencesEdit

Footnotes

  1. ^ "For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier – three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven morae [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" [14]

Citations

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Works citedEdit

External linksEdit