In mathematics, the Fibonacci numbers are the numbers in the following sequence:
By definition, the first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s.
In mathematical terms, the sequence F_{n} of Fibonacci numbers is defined by the recurrence relation
with seed values
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonaccio, "son of Bonaccio"). Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.^{[2]}^{[3]}
The Fibonacci sequence was well known in ancient India, where it was applied to the metrical sciences (prosody), before it was known in Europe. Developments have been attributed to Pingala (200 BC), Virahanka (6th century AD), Gopāla (c.1135 AD), and Hemachandra (c.1150 AD).^{[4]}
The motivation came from Sanskrit prosody, where long syllables have duration 2 and short syllables have duration 1. Any pattern of duration n can be formed by adding a short syllable to a pattern of duration n − 1, or a long syllable to a pattern of duration n − 2; thus the prosodists showed that the number of patterns of duration n is the sum of the two previous numbers in the sequence. Later authors gave algorithms for ranking and unranking these patterns (e.g. finding the kth pattern of duration n), and discovered the higherorder Fibonacci numbers. Donald Knuth reviews this work in The Art of Computer Programming.^{[5]}^{[6]}
In the West, the sequence was studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202).^{[7]} He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that: a newlyborn pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
At the end of the nth month, the number of new pairs of rabbits is equal to the number of pairs in month n2 plus the number of pairs alive last month. This is the nth Fibonacci number.^{[8]}
The first 21 Fibonacci numbers (sequence A000045 in OEIS), also denoted as F_{n}, for n = 0, 1, 2, ... ,20 are:^{[9]}^{[10]}
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 
Using the recurrence relation, the sequence can also be extended to negative index n. The result satisfies the equation
Thus the complete sequence is
Every 3rd number of the sequence is even and more generally, every kth 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
Like every sequence defined by linear recurrence, the Fibonacci numbers have a closedform solution. It has become known as Binet's formula, even though it was already known by Abraham de Moivre:^{[11]}
where
is the golden ratio (sequence A001622 in OEIS).
That
follows from the defining equation above.
The Fibonacci recursion
is similar to the defining equation of the golden ratio in the form
which is also known as the generating polynomial of the recursion.
Any root of the equation above satisfies and multiplying by shows:
By definition is a root of the equation, and the other root is Therefore:
and
Both and are geometric series (for n = 1, 2, 3, ...) that satisfy the Fibonacci recursion. The first series grows exponentially; the second exponentially tends to zero, with alternating signs. Because the Fibonacci recursion is linear, any linear combination of these two series will also satisfy the recursion. These linear combinations form a twodimensional linear vector space; the original Fibonacci sequence can be found in this space.
Linear combinations of series and , with coefficients a and b, can be defined by
All thusdefined series satisfy the Fibonacci recursion
Requiring that F_{a,b}(0) = 0 and F_{a,b}(1) = 1 yields and , resulting in the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore, an explicit check can be made:
and
establishing the base cases of the induction, proving that
Therefore, for any two starting values, a combination a,b can be found such that the function is the exact closed formula for the series.
Since for all , the number F(n) is the closest integer to Therefore it can be found by rounding, or in terms of the floor function:
Similarly, if you already know that the number F is a Fibonacci number, you can determine its index within the sequence by
Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost”, and concluded that the limit approaches the golden ratio .^{[12]}
This convergence does not depend on the starting values chosen, excluding 0, 0. For example, the initial values 19 and 31 generate the sequence 19, 31, 50, 81, 131, 212, 343, 555 ... etc. The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
In brief, Fibonacci numbers are approximately exponential: where the constant depends on starting values – as the remaining term in the exact formula for the Fibonacci numbers becomes exponentially close to zero as n grows. Taking the ratio yields
More formally, it must always follow from the explicit formula that for any real
because and thus
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 expression is also true for if the Fibonacci sequence is extended to negative integers using the Fibonacci rule
A 2dimensional system of linear difference equations that describes the Fibonacci sequence is
or
The eigenvalues of the matrix A are and , and the elements of the eigenvectors of A, and , are in the ratios and Using these facts, and the properties of eigenvalues, we can derive a direct formula for the nth element in the fibonacci series:
This matrix has a determinant of −1, and thus it is a 2×2 unimodular matrix. This property can be understood in terms of the continued fraction representation for the golden ratio:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The matrix representation gives the following closed expression for the Fibonacci numbers:
Taking the determinant of both sides of this equation yields Cassini's identity
Additionally, since A^{n}A^{m} = A^{m + n} for any square matrix A, the following identities can be derived:
In particular, with m = n,
For another way to derive the F_{2n + k} formulas see the "EWD note" by Dijkstra.^{[13]}
The question may arise whether a positive integer z is a Fibonacci number. Since F(n) is the closest integer to , the most straightforward, bruteforce test is the identity
which is true if and only if z is a Fibonacci number. In this formula, F(n) can be computed rapidly using any of the previously discussed closedform expressions.
Alternatively, a positive integer z is a Fibonacci number if and only if one of 5z^{2} + 4 or 5z^{2} − 4 is a perfect square.^{[14]}
A slightly more sophisticated test uses the fact that the convergents of the continued fraction representation of are ratios of successive Fibonacci numbers, that is the inequality
(with coprime positive integers 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 closed interval
contains a positive integer.^{[15]} The integer that the interval contains, should z be a Fibonacci number, should be the next successive Fibonacci number after z.
Most identities involving Fibonacci numbers draw from combinatorial arguments. F(n) can be interpreted as the number of sequences of 1s and 2s that sum 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. This is discussed in further detail at Young–Fibonacci lattice.
We must establish that the sequence of numbers defined by the combinatorial interpretation above satisfy the same recurrence relation as the Fibonacci numbers (and so are indeed identical to the Fibonacci numbers).
The set of F(n + 1) ways of making ordered sums of 1s and 2s that sum to n may be divided into two nonoverlapping sets. The first set contains those sums whose first summand is 1; the remainder sums to n − 1, so there are F(n) sums in the first set. The second set contains those sums whose first summand is 2; the remainder sums to n − 2, so there are F(n − 1) sums in the second set. The first summand can only be 1 or 2, so these two sets exhaust the original set. Thus F(n + 1) = F(n) + F(n−1).
We count the number of ways summing 1s and 2s to n + 1 such that at least one of the summands is 2.
As before, there are F(n + 2) ways summing 1s and 2s 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 1s, 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.
By induction:
This identity has slightly different forms for F_{j}, depending on whether j is odd or even.
The sum of the first n − 1 Fibonacci numbers, F_{j}, such that j is odd, is the (2n)th Fibonacci number.
The sum of the first n Fibonacci numbers, F_{j}, such that j is even, is the (2n + 1)th Fibonacci number minus 1.
^{[16]}
By induction for F_{2n}:
A basis case for this could be F_{1} = F_{2}.
By induction for F_{2n+1}:
A basis case for this could be F_{0} = F_{1} − 1.
By using identity 1 we can construct a telescoping sum:
If the summands are the Fibonacci numbers with even index, the proof is very similar. Summing both cases yields identity 2.
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
On the other hand, we observe from the second identity that there are
......
Adding up all n + 1 sums, we see that there are
Since the two methods of counting refer to the same number, we have
Finally, we complete the proof by subtracting the above identity from n + 1 times the second identity.
^{[11]}
Another identity useful for calculating F_{n} for large values of n is
^{[11]}
from which other identities for specific values of k, n, and c can be derived below, including
for all integers n and k. Dijkstra^{[13]} points out that doubling identities of this type can be used to calculate F_{n} using O(log n) long multiplication operations of size n bits. The number of bits of precision needed to perform each multiplication doubles at each step, so the performance is limited by the final multiplication; if the fast SchönhageStrassen multiplication algorithm is used, this is O(n log n log log n) bit operations. Notice that, with the definition of Fibonacci numbers with negative n given in the introduction, this formula reduces to the double n formula when k = 0.
Other identities include relationships to the Lucas numbers, which have the same recursive properties but start with L_{0} = 2 and L_{1} = 1. These properties include F_{2n} = F_{n}L_{n}.
There are also scaling identities, which take you from F_{n} and F_{n+1} to a variety of things of the form F_{an+b}; for instance
These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number. Such relations exist in a very general sense for numbers defined by recurrence relations. See the section on multiplication formulae under Perrin numbers for details.
The generating function of the Fibonacci sequence is the power series
This series has a simple and interesting closedform solution for
This solution can be proven by using the Fibonacci recurrence to expand each coefficient in the infinite sum defining s(x):
Solving the equation s(x) = x + xs(x) + x^{2}s(x) for s(x) results in the closed form solution.
In particular, math puzzlebooks note the curious value ,^{[17]} or more generally
for all integers k > = 0.
Conversely,
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every oddindexed reciprocal Fibonacci number as
and the sum of squared reciprocal Fibonacci numbers as
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
and there is a nice nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,
Results such as these make it plausible that a closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but none is yet known. Despite that, the reciprocal Fibonacci constant
has been proved irrational by Richard AndréJeannin.
Millin series gives a remarkable identity:^{[18]}
which follows from the closed form for its partial sums as N tends to infinity:
A Fibonacci prime is a Fibonacci number that is prime (sequence A005478 in OEIS). The first few are:
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.^{[19]} They must all have a prime index, except F_{4} = 3.
F_{kn} is divisible by F_{n}, so because there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
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).^{[20]}
144 is the only nontrivial square Fibonacci number.^{[21]} Attila Pethő proved^{[22]} in 2001 that there are only finitely many perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that only 8 and 144 are nontrivial perfect powers.^{[23]}
No Fibonacci number greater than F_{6} = 8 is one greater or one less than a prime number.^{[24]}
Any three consecutive Fibonacci numbers, taken two at a time, are relatively prime: that is,
More generally,
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 ^{[27]}^{[28]}
For example,
It is not known whether there exists a prime p such that . Such primes (if there are any) would be called WallSunSun primes.
Also, if p ≠ 5 is an odd prime number then:^{[29]}
Examples of all the cases:
For odd n, all odd prime divisors of F_{n} are ≡ 1 (mod 4), implying that all odd divisors of F_{n} (as the products of odd prime divisors) are ≡ 1 (mod 4).^{[30]}^{[31]}
For example, F_{1} = 1, F_{3} = 2, F_{5} = 5, F_{7} = 13, F_{9} = 34 = 2×17, F_{11} = 89, F_{13} = 233, F_{15} = 610 = 2×5×61
For example, let n = 1:
F_{1} + F_{2} + ... + F_{10} = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11×13
n = 2:
F_{2}+F_{3}+...+F_{11} = 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 = 231 = 11×21
n = 3:
F_{3}+F_{4}+...+F_{12} = 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144= 374 = 11×34
In fact, the identity is true for all integers n, not just positive ones:
n = 0:
F_{0}+F_{1}+...+F_{9} = 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 = 88 = 11×8
n = −1:
F_{−1}+F_{0}+...+F_{8} = 1 + 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 = 55 = 11×5
n = −2:
F_{−2}+F_{−1}+F_{0}+...+F_{7} = −1 + 1 + 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 = 33 = 11×3
It may be seen that if the members of the Fibonacci sequence are taken mod n, the resulting sequence must be periodic with period at most n^{2}. The lengths of the periods for various n form the socalled Pisano periods (sequence A001175 in OEIS). Determining the Pisano periods in general is an open problem,^{[citation needed]} although for any particular n it can be solved as an instance of cycle detection.
Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.
The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely. The triangle sides a, b, c can be calculated directly:
These formulas satisfy for all n, but they only represent triangle sides when n > 2.
Any four consecutive Fibonacci numbers F_{n}, F_{n+1}, F_{n+2} and F_{n+3} can also be used to generate a Pythagorean triple in a different way:
Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then:
Example 2: let the Fibonacci numbers be 8, 13, 21 and 34. Then:
Since F_{n} is asymptotic to , the number of digits in 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 F_{n} is asymptotic to .
The Fibonacci numbers are important in the runtime analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.^{[32]}
Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.
The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle and Lozanić's triangle (see Binomial coefficient). They occur more obviously in Hosoya's triangle.
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.
The Fibonacci numbers and principle is also used in the financial markets. It is used in trading algorithms, applications and strategies. Some typical forms include: the Fibonacci fan, the Fibonacci arc, Fibonacci retracement and the Fibonacci time extension.^{[citation needed]}
Fibonacci numbers are used by some pseudorandom number generators.
Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers  by dividing the list so that the two parts have lengths in the approximate proportion φ. A tapedrive implementation of the polyphase merge sort was described in The Art of Computer Programming.
Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.
The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.
A onedimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers.^{[33]}
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 e.g. µlaw.^{[34]}^{[35]}
In music, Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements. It is commonly thought that the first movement of Béla Bartók's Music for Strings, Percussion, and Celesta was structured using Fibonacci numbers.
Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio (denoted φ), the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.^{[36]}^{[37]}^{[38]}
Fibonacci sequences appear in biological settings,^{[39]} in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,^{[40]} the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.^{[41]} In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves.^{[42]} The Fibonacci numbers are also found in the family tree of honeybees.^{[43]}
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.^{[44]}
A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.^{[45]} 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. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.^{[46]}
Fibonacci numbers also appear in the description of the reproduction of a population of idealized bees, according to the following rules:
Thus, a male bee will always have one parent, and a female bee will have two.
If one traces the ancestry of any male bee (1 bee), he has 1 female parent (1 bee). This female had 2 parents, a male and a female (2 bees). The female had two parents, a male and a female, and the male had one female (3 bees). Those two females each had two parents, and the male had one (5 bees). This sequence of numbers of parents is the Fibonacci sequence.^{[47]}
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.
The Fibonacci sequence has been generalized in many ways. These include:
