In mathematics, the **Fibonacci numbers**, commonly denoted *F _{n}*, form a sequence, called the

- History
- Sequence properties
- Relation to the golden ratio
- Closed-form expression
- Computation by rounding
- Limit of consecutive quotients
- Decomposition of powers
- Matrix form
- Identification
- Combinatorial identities
- Combinatorial proofs
- Symbolic method
- Induction proofs
- Binet formula proofs
- Other identities
- Cassini's and Catalan's identities
- d'Ocagne's identity
- Generating function
- Reciprocal sums
- Primes and divisibility
- Divisibility properties
- Primality testing
- Fibonacci primes
- Prime divisors
- Periodicity modulo n
- Magnitude
- Generalizations
- Applications
- Mathematics
- Computer science
- Nature
- Other
- See also
- References
- Works cited
- External links

and

for *n* > 1.

The sequence starts:^{ [2] }

- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

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] }

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 .

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 *F*_{m + 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 (*F*_{m+1}) is obtained by adding one [S] to the *F*_{m} cases and one [L] to the *F*_{m−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].^{ [lower-alpha 1] }

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] }

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 *n*th 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 *n*th month is the *n*th Fibonacci number.^{ [20] }

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

The first 21 Fibonacci numbers *F _{n}* are:

*F*_{0}*F*_{1}*F*_{2}*F*_{3}*F*_{4}*F*_{5}*F*_{6}*F*_{7}*F*_{8}*F*_{9}*F*_{10}*F*_{11}*F*_{12}*F*_{13}*F*_{14}*F*_{15}*F*_{16}*F*_{17}*F*_{18}*F*_{19}*F*_{20}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}*F*_{0}*F*_{1}*F*_{2}*F*_{3}*F*_{4}*F*_{5}*F*_{6}*F*_{7}*F*_{8}−21 13 −8 5 −3 2 −1 1 0 1 1 2 3 5 8 13 21

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 ( OEIS: A001622 ), 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 *U*_{0} = 0 and *U*_{1} = 1 then the resulting sequence *U*_{n} 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 *U*_{0} and *U*_{1} to be arbitrary constants, a more general solution is:

where

Since

for all *n* ≥ 0, the number *F*_{n} 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

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.

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 *F _{n}* is extended to negative integers using the Fibonacci rule

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 **A**^{n}**A**^{m} = **A**^{n+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 *n*th 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] }

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).

Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that *F*_{n} 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 *F*_{n}, with the convention that *F*_{0} = 0, meaning no sum adds up to −1, and that *F*_{1} = 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 *n*th Fibonacci number is the sum of the previous two Fibonacci numbers, may be shown by dividing the *F*_{n} 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 *F*_{n-1} sums, and in the second group the remaining terms add to *n* − 3, so there are *F*_{n−2} sums. So there are a total of *F*_{n−1} + *F*_{n−2} sums altogether, showing this is equal to *F*_{n}.

Similarly, it may be shown that the sum of the first Fibonacci numbers up to the *n*th 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 *F*_{2n−1} is the (2*n*)th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to *F*_{2n} is the (2*n* + 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 *F*_{n} is the product of the *n*th and (*n* + 1)th Fibonacci numbers. In this case Fibonacci rectangle of size *F*_{n} by *F*(*n* + 1) can be decomposed into squares of size *F*_{n}, *F*_{n−1}, and so on to *F*_{1} = 1, from which the identity follows by comparing areas (see the first image at the top of this article).

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 .

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

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 required, using the facts and to simplify the equations.

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

Cassini's identity states that

Catalan's identity is a generalization:

where *L*_{n} 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.

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.

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,

The sum of all even-indexed reciprocal Fibonacci numbers is^{ [35] }

with the Lambert series because there is the equality

So the reciprocal Fibonacci constant is

Moreover, this number has been proved irrational by Richard André-Jeannin.^{ [36] }

The **Millin series** gives the identity^{ [37] }

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

Every third number of the sequence is even and more generally, every *k*th number of the sequence is a multiple of *F _{k}*. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property

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

- gcd(
*F*_{n},*F*_{n+1}) = gcd(*F*_{n},*F*_{n+2}) = gcd(*F*_{n+1},*F*_{n+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 *F*_{p − 1}, and if *p* is congruent to 2 or 3 (mod 5), then, *p* divides *F*_{p + 1}. The remaining case is that *p* = 5, and in this case *p* divides *F*_{p}.

These cases can be combined into a single, non-piecewise formula, using the Legendre symbol:^{ [40] }

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 *F*_{m} (mod *n*) efficiently using the matrix form. Thus

Here the matrix power *A*^{m} is calculated using modular exponentiation, which can be adapted to matrices.^{ [41] }

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

- 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... OEIS: A005478 .

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.^{ [42] }

*F*_{kn} is divisible by *F*_{n}, so, apart from *F*_{4} = 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 *F*_{6} = 8 is one greater or one less than a prime number.^{ [43] }

The only nontrivial square Fibonacci number is 144.^{ [44] } Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.^{ [45] } In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.^{ [46] }

1, 3, 21, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.^{ [47] }

No Fibonacci number can be a perfect number.^{ [48] } More generally, no Fibonacci number other than 1 can be multiply perfect,^{ [49] } and no ratio of two Fibonacci numbers can be perfect.^{ [50] }

With the exceptions of 1, 8 and 144 (*F*_{1} = *F*_{2}, *F*_{6} and *F*_{12}) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).^{ [51] } As a result, 8 and 144 (*F*_{6} and *F*_{12}) are the only Fibonacci numbers that are the product of other Fibonacci numbers OEIS: A235383 .

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

^{ [52] }^{ [53] }

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:^{ [54] }

**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 *F*_{n} are congruent to 1 modulo 4, implying that all odd divisors of *F*_{n} (as the products of odd prime divisors) are congruent to 1 modulo 4.^{ [55] }

For example,

All known factors of Fibonacci numbers *F*(*i*) for all *i* < 50000 are collected at the relevant repositories.^{ [56] }^{ [57] }

If the members of the Fibonacci sequence are taken mod *n*, the resulting sequence is periodic with period at most *6n*.^{ [58] } The lengths of the periods for various *n* form the so-called Pisano periods OEIS: A001175 . 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.

Since *F _{n}* is asymptotic to , the number of digits in

More generally, in the base *b* representation, the number of digits in *F*_{n} is asymptotic to

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
*L*_{1}= 1,*L*_{2}= 3, and*L*=_{n}*L*_{n−1}+*L*_{n−2}. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. - Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have
*P*= 2_{n}*P*_{n − 1}+*P*_{n − 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*.^{ [59] }

The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):^{ [60] }

The generating function can be expanded into

and collecting like terms of , we have the identity

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-k-1* terms.

These numbers also give the solution to certain enumerative problems,^{ [61] } 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 *F*_{n+1} ways to do this (equivalently, it's also the number of domino tilings of the rectangle). For example, there are *F*_{5+1} = *F*_{6} = 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.

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*F*_{n+2}. For example, out of the 16 binary strings of length 4, there are*F*_{6}= 8 without consecutive 1s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Equivalently,*F*_{n+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*F*_{n+1}. For example, out of the 16 binary strings of length 4, there are*F*_{5}= 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*F*_{n+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 2*F*_{n}. For example, out of the 16 binary strings of length 4, there are 2*F*_{4}= 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.
^{ [62] } - 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.
^{ [63] } - 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.
^{ [64] }

- 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.
^{ [65] } - 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.
^{ [66] } - 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.
^{ [67] } - 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.
^{ [68] }^{ [69] } - They are also used in planning poker, which is a step in estimating in software development projects that use the Scrum methodology.

Fibonacci sequences appear in biological settings,^{ [70] } such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,^{ [71] } the flowering of artichoke, an uncurling fern and the arrangement of a pine cone,^{ [72] } and the family tree of honeybees.^{ [73] }^{ [74] } Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.^{ [75] } Field daisies most often have petals in counts of Fibonacci numbers.^{ [76] } In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.^{ [77] }

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.^{ [78] }

A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel in 1979.^{ [79] } 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,^{ [80] } typically counted by the outermost range of radii.^{ [81] }

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, *F*_{n}, is the number of female ancestors, which is *F*_{n−1}, plus the number of male ancestors, which is *F*_{n−2}.^{ [82] } This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

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.^{ [83] } 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.^{ [84] }

- 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.)^{ [85] } - 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.^{ [86] } - Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics.
^{ [87] }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.
^{ [88] } - 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.
^{ [89] }See also Golden ratio § Music.

In number theory, **Euler's totient function** counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called **Euler's phi function**. In other words, it is the number of integers k in the range 1 ≤ *k* ≤ *n* for which the greatest common divisor gcd(*n*, *k*) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.

In mathematics, the **Hermite polynomials** are a classical orthogonal polynomial sequence.

In number theory, the study of **Diophantine approximation** deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

In mathematics, a **Green's function** is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.

In mathematics, the **Farey sequence** of order *n* is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to *n*, arranged in order of increasing size.

The **Lucas numbers** or **Lucas series** are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.

The **path integral formulation** is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

The **Gram–Charlier A series**, and the **Edgeworth series** are series that approximate a probability distribution in terms of its cumulants. The series are the same; but, the arrangement of terms differ. The key idea of these expansions is to write the characteristic function of the distribution whose probability density function f is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover f through the inverse Fourier transform.

In quantum field theory, the **LSZ reduction formula** is a method to calculate *S*-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.

The following are important identities involving derivatives and integrals in vector calculus.

In functional analysis and quantum measurement theory, a **positive operator-valued measure** (**POVM**) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs.

In mathematics, the **Chebyshev function** is either of two related functions. The **first Chebyshev function***ϑ*(*x*) or *θ*(*x*) is given by

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

The **square root of 5** is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the **principal square root of 5**, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

The **Leonardo numbers** are a sequence of numbers given by the recurrence:

In mathematics, **vector spherical harmonics** (**VSH**) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

A **helium atom** is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with either one or two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom.

In mathematics, **infinite compositions of analytic functions (ICAF)** offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a *single function* see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.

In mathematics, two quantities are in the **supergolden ratio** if the quotient of the larger number divided by the smaller one is equal to

In mathematics, a **conical spiral** is a curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called conchospiral.

**Footnotes**

- ↑ "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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*File:Liber abbaci magliab f124r.jpg - Wikimedia Commons*. - 1 2 Pisano 2002, pp. 404–05.
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*Toward a Global Science*, Indiana University Press, p. 126, ISBN 978-0-253-33388-9 - 1 2 Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India",
*Historia Mathematica*,**12**(3): 229–44, doi: 10.1016/0315-0860(85)90021-7 - 1 2 Knuth, Donald (2006),
*The Art of Computer Programming*, 4. Generating All Trees – History of Combinatorial Generation, Addison–Wesley, p. 50, ISBN 978-0-321-33570-8,it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ...there are exactly Fm+1 of them. For example the 21 sequences when

*m*= 7 are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1) - ↑ Knuth, Donald (1968),
*The Art of Computer Programming*,**1**, Addison Wesley, p. 100, ISBN 978-81-7758-754-8,Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)...

- 1 2 Livio 2003, p. 197.
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*Pāṇinikālīna Bhāratavarṣa**(Hn.). Varanasi-I: TheChowkhamba Vidyabhawan*,SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala [1969, 463–76], after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC

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