thumb|300px|right|A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21.
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones.
The sequence commonly starts from 0 and 1, although some authors omit the initial terms and start the sequence from 1 and 1 or from 1 and 2.
Starting from 0 and 1, the next few values in the sequence are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.
They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.
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 strongly related to the golden ratio: Binet's formula expresses the th Fibonacci number in terms of  and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as  increases.
Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.
Definition
The Fibonacci numbers may be defined by the recurrence relation  F_0=0,\quad F_1= 1, and F_n=F_{n-1} + F_{n-2} for .
Under some older definitions, the value F_0 = 0 is omitted, so that the sequence starts with F_1=F_2=1, and the recurrence F_n=F_{n-1} + F_{n-2} is valid for .
Fibonacci started the sequence with F_1=1, F_2=2.
The first 21 Fibonacci numbers  are:
{| class="wikitable" style="text-align:right"
History
thumb|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.
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  units is .
Knowledge of the Fibonacci sequence was expressed as early as Pingala ( 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  beats () is obtained by adding one [S] to the  cases and one [L] to the  cases.
Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).
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):
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].
Hemachandra (c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."
thumb|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 where it is used to calculate the growth of rabbit populations.
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 th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month ) plus the number of pairs alive last month (month ).
The number in the th month is the th Fibonacci number.
The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.
Relation to the golden ratio
{{anchor|Binet's formula}}Closed-form expression
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:
F_n = \frac{\varphi^n-\psi^n}{\varphi-\psi} = \frac{\varphi^n-\psi^n}{\sqrt 5}, where \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\ldots is the golden ratio (), and \psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = - {1 \over \varphi} \approx -0.61803\,39887\ldots.
Since \psi = -\varphi^{-1}, this formula can also be written as F_n = \frac{\varphi^n-(-\varphi)^{-n}}{\sqrt 5} = \frac{\varphi^n-(-\varphi)^{-n}}{2 \varphi - 1}.
To see this, note that  and  are both solutions of the equations x^2 = x + 1 \quad\text{and}\quad x^n = x^{n-1} + x^{n-2}, so the powers of  and  satisfy the Fibonacci recursion.
In other words, \varphi^n  = \varphi^{n-1} + \varphi^{n-2} and \psi^n = \psi^{n-1} + \psi^{n-2}.
It follows that for any values  and , the sequence defined by U_n=a \varphi^n + b \psi^n satisfies the same recurrence.
U_{n-1} + U_{n-2} = a\varphi^{n-1} + b\psi^{n-1} + a\varphi^{n-2} + b\psi^{n-2} = a\varphi^{n-1} + a\varphi^{n-2} +  b\psi^{n-1} + b\psi^{n-2} = U_n
If  and  are chosen so that  and  then the resulting sequence  must be the Fibonacci sequence.
This is the same as requiring  and  satisfy the system of equations: \left\{\begin{array}{l} a + b = 0\\ \varphi a + \psi b = 1\end{array}\right.
which has solution a = \frac{1}{\varphi-\psi} = \frac{1}{\sqrt 5},\quad  b = -a, producing the required formula.
Taking the starting values  and  to be arbitrary constants, a more general solution is:  U_n=a\varphi^n+b\psi^n  where  a=\frac{U_1-U_0\psi}{\sqrt 5}  b=\frac{U_0\varphi-U_1}{\sqrt 5}.
Computation by rounding
Since \left|\frac{\psi^{n}}{\sqrt 5}\right| < \frac{1}{2}
for all , the number  is the closest integer to \frac{\varphi^n}{\sqrt 5}.
Therefore, it can be found by rounding, using the nearest integer function: F_n=\left[\frac{\varphi^n}{\sqrt 5}\right],\ n \geq 0.
In fact, the rounding error is very small, being less than 0.1 for , and less than 0.01 for .
Fibonacci numbers can also be computed by truncation, in terms of the floor function: F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\right\rfloor,\ n \geq 0.
As the floor function is monotonic, the latter formula can be inverted for finding the index   of the largest Fibonacci number that is not greater than a real number : n(F) = \left\lfloor \log_\varphi \left(F\cdot\sqrt{5} + \frac{1}{2}\right) \right\rfloor, where \log_\varphi(x) = \ln(x)/\ln(\varphi) = \log_{10}(x)/\log_{10}(\varphi).
Magnitude
Since Fn is asymptotic to \varphi^n/\sqrt5, the number of digits in Fn is asymptotic to n\log_{10}\varphi\approx 0.2090\, n.
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 n\log_b\varphi.
Limit of consecutive quotients
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 \varphi\colon   \lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.
This convergence holds regardless of the starting values, excluding 0 and 0, or any pair in the conjugate golden ratio, -1/\varphi.
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.
In general, \lim_{n\to\infty}\frac{F_{n+m}}{F_n}=\varphi^m , because the ratios between consecutive Fibonacci numbers approaches \varphi.
thumb|upright=2.2|left|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 powers
Since the golden ratio satisfies the equation \varphi^2 = \varphi + 1,
this expression can be used to decompose higher powers \varphi^n as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of \varphi and 1.
The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients: \varphi^n = F_n\varphi + F_{n-1}.
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 F_n = F_{n-1} + F_{n-2}.
Identification
Binet's formula provides a proof that a positive integer x is a Fibonacci number if and only if at least one of 5x^2+4 or 5x^2-4 is a perfect square.
This is because Binet's formula above can be rearranged to give
n = \log_{\varphi}\left(\frac{F_n\sqrt{5} + \sqrt{5{F_n}^2 \pm 4}}{2}\right),
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).
Matrix form
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
{F_{k+2} \choose F_{k+1}} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k}}  alternatively denoted  \vec F_{k+1} = \mathbf{A} \vec F_{k},
which yields \vec F_{n} = \mathbf{A}^n  \vec F_{0}.
The eigenvalues of the matrix  are \varphi=\frac12(1+\sqrt5) and -\varphi^{-1}=\frac12(1-\sqrt5) corresponding to the respective eigenvectors  \vec \mu={\varphi \choose 1}  and \vec\nu={-\varphi^{-1} \choose 1}.
As the initial value is \vec F_0={1 \choose 0}=\frac{1}{\sqrt{5}}\vec{\mu}-\frac{1}{\sqrt{5}}\vec{\nu},  it follows that the th term is \begin{align}\vec F_{n} &= \frac{1}{\sqrt{5}}A^n\vec\mu-\frac{1}{\sqrt{5}}A^n\vec\nu \\ &= \frac{1}{\sqrt{5}}\varphi^n\vec\mu-\frac{1}{\sqrt{5}}(-\varphi)^{-n}\vec\nu~\\ & =\cfrac{1}{\sqrt{5}}\left(\cfrac{1+\sqrt{5}}{2}\right)^n{\varphi \choose 1}-\cfrac{1}{\sqrt{5}}\left(\cfrac{1-\sqrt{5}}{2}\right)^n{-\varphi^{-1}\choose 1}, \end{align} From this, the th element in the Fibonacci series may be read off directly as a closed-form expression: F_{n} = \cfrac{1}{\sqrt{5}}\left(\cfrac{1+\sqrt{5}}{2}\right)^n-\cfrac{1}{\sqrt{5}}\left(\cfrac{1-\sqrt{5}}{2}\right)^n.
Equivalently, the same computation may performed by diagonalization of  through use of its eigendecomposition: \begin{align} A & = S\Lambda S^{-1} ,\\  A^n & = S\Lambda^n S^{-1}, \end{align} where \Lambda=\begin{pmatrix} \varphi & 0 \\ 0 & -\varphi^{-1} \end{pmatrix} and S=\begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}.
The closed-form expression for the th element in the Fibonacci series is therefore given by
\begin{align} {F_{n+1} \choose F_{n}} & = A^{n} {F_1 \choose F_0} \\  & = S \Lambda^n S^{-1} {F_1 \choose F_0} \\  & = S \begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix} S^{-1} {F_1 \choose F_0} \\  & = \begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}      \begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix}      \frac{1}{\sqrt{5}}\begin{pmatrix} 1 & \varphi^{-1} \\ -1 & \varphi \end{pmatrix} {1 \choose 0}, \end{align}
which again yields F_{n} = \cfrac{\varphi^n-(-\varphi)^{-n}}{\sqrt{5}}.
The 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:
\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}.
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-form expression for the Fibonacci numbers:
\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}.
Taking the determinant of both sides of this equation yields Cassini's identity, (-1)^n = F_{n+1}F_{n-1} - {F_n}^2.
Moreover, since  for any square matrix , 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  into ), \begin{align}  {F_m}{F_n} + {F_{m-1}}{F_{n-1}} &= F_{m+n-1},\\  F_{m} F_{n+1} + F_{m-1} F_n &= F_{m+n}  .
\end{align}
In particular, with , \begin{array}{ll}  F_{2 n-1} &= {F_n}^2 + {F_{n-1}}^2\\  F_{2 n}   &= (F_{n-1}+F_{n+1})F_n\\           &= (2 F_{n-1}+F_n)F_n\\           &= (2 F_{n+1}-F_n)F_n. \end{array}
These last two identities provide a way to compute Fibonacci numbers recursively in  arithmetic operations and in time , where  is the time for the multiplication of two numbers of  digits.
This matches the time for computing the 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).
Combinatorial identities
Combinatorial proofs
Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that F_n can be interpreted as the number of [possibly empty] sequences of 1s and 2s whose sum is n-1.
This can be taken as the definition of F_{n} with the conventions F_{0}=0, meaning no such sequence exists whose sum is −1, and F_{1}=1, meaning the empty sequence "adds up" to 0.
In the following, |{...}| is the cardinality of a set:
F_{0} = 0 = |\{\}|
F_{1} = 1 = |\{\{\}\}|
F_{2} = 1 = |\{\{1\}\}|
F_{3} = 2 = |\{\{1,1\},\{2\}\}|
F_{4} = 3 = |\{\{1,1,1\},\{1,2\},\{2,1\}\}|
F_{5} = 5 = |\{\{1,1,1,1\},\{1,1,2\},\{1,2,1\},\{2,1,1\},\{2,2\}\}|
In this manner the recurrence relation F_{n} = F_{n-1} + F_{n-2} may be understood by dividing the F_{n} sequences into two non-overlapping sets where all sequences either begin with 1 or 2: F_{n} = |\{\{1,...\},\{1,...\},...\}| + |\{\{2,...\},\{2,...\},...\}| Excluding the first element, the remaining terms in each sequence sum to n-2 or n-3 and the cardinality of each set is F_{n-1} or F_{n-2} giving a total of F_{n-1}+F_{n-2} sequences, showing this is equal to F_{n}.
In a similar manner 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.
In symbols: \sum_{i=1}^n F_i = F_{n+2} - 1
This may be seen by dividing all sequences summing to n+1 based on the location of the first 2.
Specifically, each set consists of those sequences that start \{2,...\}, \{1,2,...\}, ...,  until the last two sets \{\{1,1,...,1,2\}\}, \{\{1,1,...,1\}\} each with cardinality 1.
Following the same logic as before, by summing the cardinality of each set we see that
F_{n+2} = F_{n} + F_{n-1} + ... + |\{\{1,1,...,1,2\}\}| + |\{\{1,1,...,1\}\}|
... where the last two terms have the value F_{1} = 1.
From this it follows that \sum_{i=1}^n F_i = F_{n+2}-1.
A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities: \sum_{i=0}^{n-1} F_{2 i+1} = F_{2 n} and \sum_{i=1}^{n} F_{2 i} = F_{2 n+1}-1.
In words, the sum of the first Fibonacci numbers with odd index up to F_{2 n-1} is the (2n)th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to F_{2 n} is the (2n + 1)th Fibonacci number minus 1.
A different trick may be used to prove \sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1} or in words, the sum of the squares of the first Fibonacci numbers up to F_{n} is the product of the nth and (n + 1)th Fibonacci numbers.
To see this, begin with a Fibonacci rectangle of size F_{n} \times F_{n+1} and decompose it into squares of size F_{n}, F_{n-1}, ..., F_{1}; from this the identity follows by comparing areas: thumb|center Symbolic method
The sequence (F_n)_{n\in\mathbb N} is also considered using the symbolic method.
More precisely, this sequence corresponds to a specifiable combinatorial class.
The specification of this sequence is \operatorname{Seq}(\mathcal{Z+Z^2}).
Indeed, as stated above, the n-th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of n-1 using terms 1 and 2.
It follows that the ordinary generating function of the Fibonacci sequence, i.e. \sum_{i=0}^\infty F_iz^i, is the complex function \frac{z}{1-z-z^2}.
Induction proofs
Fibonacci identities often can be easily proved used mathematical induction.
For example, reconsider \sum_{i=1}^n F_i = F_{n+2} - 1.
Adding F_{n+1} to both sides gives
\sum_{i=1}^n F_i + F_{n+1} = F_{n+1} + F_{n+2} - 1
and so we have the formula for n+1 \sum_{i=1}^{n+1} F_i = F_{n+3} - 1
Similarly, add F^2_{n+1} to both sides of \sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1} to give \sum_{i=1}^n {F_i}^2 + F^2_{n+1} = F_{n+1}\left(F_{n} + F_{n+1}\right) \sum_{i=1}^{n+1} {F_i}^2 = F_{n+1}F_{n+2} Binet formula proofs
The Binet formula is \sqrt5F_n = \varphi^n - \psi^n.
This can be used to prove Fibonacci identities.
For example, to prove that \sum_{i=1}^n F_i = F_{n+2} - 1 note that the left hand side multiplied by \sqrt5 becomes
\begin{align} 1 +& \varphi + \varphi^2 + \dots + \varphi^n - \left(1 + \psi + \psi^2 + \dots + \psi^n \right)\\ &= \frac{\varphi^{n+1}-1}{\varphi-1} - \frac{\psi^{n+1}-1}{\psi-1}\\ &= \frac{\varphi^{n+1}-1}{-\psi} - \frac{\psi^{n+1}-1}{-\varphi}\\ &= \frac{-\varphi^{n+2}+\varphi + \psi^{n+2}-\psi}{\varphi\psi}\\ &= \varphi^{n+2}-\psi^{n+2}-(\varphi-\psi)\\ &= \sqrt5(F_{n+2}-1)\\ \end{align} as required, using the facts \varphi\psi =- 1 and \varphi-\psi=\sqrt5 to simplify the equations.
Other identities
Numerous other identities can be derived using various methods.
Here are some of them: Cassini's and Catalan's identities
Cassini's identity states that {F_n}^2 - F_{n+1}F_{n-1} = (-1)^{n-1} Catalan's identity is a generalization: {F_n}^2 - F_{n+r}F_{n-r} = (-1)^{n-r}F_r^2 d'Ocagne's identity
F_m F_{n+1} - F_{m+1} F_n = (-1)^n F_{m-n} F_{2 n} = {F_{n+1}}^2 - {F_{n-1}}^2 = F_n \left (F_{n+1}+F_{n-1} \right ) = F_nL_n where Ln is the n'th Lucas number.
The last is an identity for doubling n; other identities of this type are F_{3 n} = 2{F_n^3} + 3 F_n F_{n+1} F_{n-1} = 5{F_n}^3 + 3 (-1)^n F_n by Cassini's identity.
F_{3 n+1} = F_{n+1}^3 + 3 F_{n+1}{F_n}^2 - F_n^3 F_{3 n+2} = F_{n+1}^3 + 3 F_{n+1}^2 F_n + {F_n}^3 F_{4 n} = 4 F_n F_{n+1} \left ({F_{n+1}}^2 + 2{F_n}^2 \right ) - 3{F_n}^2 \left ({F_n}^2 + 2{F_{n+1}}^2 \right )
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,
F_{k n+c} = \sum_{i=0}^k {k\choose i} F_{c-i} F_n^i F_{n+1}^{k-i}.
or alternatively
F_{k n+c} = \sum_{i=0}^k {k\choose i} F_{c+i} F_n^i F_{n-1}^{k-i}.
Putting  in this formula, one gets again the formulas of the end of above section Matrix form.
Generating function
The generating function of the Fibonacci sequence is the power series s(x)=\sum_{k=0}^{\infty} F_k x^k = \sum_{k=1}^{\infty} F_k x^k = 0+x+x^2+2 x^3+3 x^4+\dots.
This series is convergent for |x| < \frac{1}{\varphi}, and its sum has a simple closed-form: s(x)=\frac{x}{1-x-x^2}
This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum: \begin{align}   s(x) &= \sum_{k=0}^{\infty} F_k x^k \\        &= F_0 + F_1x + \sum_{k=2}^{\infty} F_k x^k \\        &= F_0 + F_1x + \sum_{k=2}^{\infty} \left( F_{k-1} + F_{k-2} \right) x^k \\        &= x + \sum_{k=2}^{\infty} F_{k-1} x^k + \sum_{k=2}^{\infty} F_{k-2} x^k \\        &= x + x\sum_{k=2}^{\infty} F_{k-1} x^{k-1} + x^2\sum_{k=2}^{\infty} F_{k-2} x^{k-2} \\        &= x + x\sum_{k=1}^{\infty} F_k x^k + x^2\sum_{k=0}^{\infty} F_k x^k \\        &= x + x s(x) + x^2 s(x).
\end{align}
Solving the equation s(x)=x+xs(x)+x^2 s(x) for s(x) results in the closed form.
-s\left(-\frac{1}{x}\right) gives the generating function for the negafibonacci numbers, and s(x) satisfies the functional equation s(x)=s\left(-\frac{1}{x}\right).
The partial fraction decomposition is given by s(x) = \frac{1}{\sqrt5}\left(\frac{1}{1 - \varphi x} - \frac{1}{1 - \psi x}\right) where \varphi = \frac{1 + \sqrt{5}}{2} is the golden ratio and \psi = \frac{1 - \sqrt{5}}{2} is its conjugate.
Reciprocal sums
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions.
For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as  \sum_{k=1}^\infty \frac{1}{F_{2 k-1}} = \frac{\sqrt{5}}{4} \;\, \vartheta_2\!
\left(0, \frac{3-\sqrt 5}{2}\right)^2 ,
and the sum of squared reciprocal Fibonacci numbers as \sum_{k=1}^\infty \frac{1}{{F_k}^2} = \frac{5}{24} \left(\vartheta_2\!\left(0, \frac{3-\sqrt 5}{2}\right)^4 - \vartheta_4\!\left(0, \frac{3-\sqrt 5}{2}\right)^4 + 1 \right).
If we add 1 to each Fibonacci number in the first sum, there is also the closed form \sum_{k=1}^\infty \frac{1}{1+F_{2 k-1}} = \frac{\sqrt{5}}{2},
and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio, \sum_{k=1}^\infty \frac{(-1)^{k+1}}{\sum_{j=1}^k {F_{j}}^2} = \frac{\sqrt{5}-1}{2} .
The sum of all even-indexed reciprocal Fibonacci numbers isLandau (1899) quoted according Borwein, Page 95, Exercise 3b.
\sum_{k=1}^{\infty} \frac{1}{F_{2 k}} = \sqrt{5} \left(L\bigl(\psi^2\bigr) - L\bigl(\psi^4\bigr)\right)  with the Lambert series   \textstyle L(q) := \sum_{k=1}^{\infty} \frac{q^k}{1-q^k} ,   since   \textstyle \frac{1}{F_{2 k}} = \sqrt{5} \left(\frac{\psi^{2 k}}{1-\psi^{2 k}} - \frac{\psi^{4 k}}{1-\psi^{4 k}} \right).
So the reciprocal Fibonacci constant is \sum_{k=1}^{\infty} \frac{1}{F_k} = \sum_{k=1}^\infty \frac{1}{F_{2 k-1}} + \sum_{k=1}^{\infty} \frac {1}{F_{2 k}} = 3.359885666243 \dots
Moreover, this number has been proved irrational by Richard André-Jeannin.
The Millin series gives the identity \sum_{k=0}^{\infty} \frac{1}{F_{2^k}} = \frac{7 - \sqrt{5}}{2}, which follows from the closed form for its partial sums as N tends to infinity: \sum_{k=0}^N \frac{1}{F_{2^k}} = 3 - \frac{F_{2^N-1}}{F_{2^N}}.
Primes and divisibility
Divisibility properties
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 \gcd(F_m,F_n) = F_{\gcd(m,n)}.
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.
\begin{cases} p =5 & \Rightarrow p \mid F_{p}, \\ p \equiv \pm1 \pmod 5 & \Rightarrow p \mid F_{p-1}, \\ p \equiv \pm2 \pmod 5 &  \Rightarrow p \mid F_{p+1}.
\end{cases}
These cases can be combined into a single, non-piecewise formula, using the Legendre symbol:.
Williams calls this property "well known".
p \mid F_{p-\left(\frac{5}{p}\right)}.
Primality testing
The above formula can be used as a primality test in the sense that if n \mid F_{n-\left(\frac{5}{n}\right)}, 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 largesay a 500-bit numberthen we can calculate Fm (mod n) efficiently using the matrix form.
Thus
\begin{pmatrix} F_{m+1} & F_m \\ F_m & F_{m-1} \end{pmatrix} \equiv \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^m \pmod n.
Here the matrix power Am is calculated using modular exponentiation, which can be adapted to matrices.Prime Numbers, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.
Fibonacci primes
A Fibonacci prime is a Fibonacci number that is prime.
The first few are:
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... .
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.
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.
The only nontrivial square Fibonacci number is 144.
Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.
In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.
1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.
No Fibonacci number can be a perfect number.
More generally, no Fibonacci number other than 1 can be multiply perfect, and no ratio of two Fibonacci numbers can be perfect.
Prime divisors
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).
As a result, 8 and 144 (F6 and F12) are the only Fibonacci numbers that are the product of other Fibonacci numbers .
The divisibility of Fibonacci numbers by a prime p is related to the Legendre symbol \left(\tfrac{p}{5}\right) which is evaluated as follows: \left(\frac{p}{5}\right) = \begin{cases} 0 & \text{if } p = 5\\ 1 & \text{if } p \equiv \pm 1 \pmod 5\\ -1 & \text{if } p \equiv \pm 2 \pmod 5.
\end{cases}
If p is a prime number then  F_p \equiv \left(\frac{p}{5}\right) \pmod p \quad \text{and}\quad F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p.
For example, \begin{align} (\tfrac{2}{5}) &= -1, &F_3  &= 2, &F_2&=1, \\ (\tfrac{3}{5}) &= -1, &F_4  &= 3,&F_3&=2, \\ (\tfrac{5}{5}) &= 0, &F_5  &= 5, \\ (\tfrac{7}{5}) &= -1, &F_8  &= 21,&F_7&=13, \\ (\tfrac{11}{5})& = +1, &F_{10}&  = 55, &F_{11}&=89.
\end{align}
It is not known whether there exists a prime p such that
F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod{p^2}.
Such primes (if there are any) would be called Wall–Sun–Sun primes.
Also, if p ≠  5 is an odd prime number then: 5 F^2_{\frac{p \pm 1}{2}} \equiv \begin{cases} \tfrac{1}{2} \left (5\left(\frac{p}{5}\right)\pm 5 \right ) \pmod p & \text{if } p \equiv 1 \pmod 4\\ \tfrac{1}{2} \left (5\left(\frac{p}{5}\right)\mp 3 \right ) \pmod p & \text{if } p \equiv 3 \pmod 4.
\end{cases}
Example 1.
p = 7, in this case p ≡ 3 (mod 4) and we have: (\tfrac{7}{5}) = -1: \qquad \tfrac{1}{2}\left (5(\tfrac{7}{5})+3 \right ) =-1, \quad \tfrac{1}{2} \left (5(\tfrac{7}{5})-3 \right )=-4.
F_3=2 \text{ and } F_4=3.
5F_3^2=20\equiv -1 \pmod {7}\;\;\text{ and }\;\;5F_4^2=45\equiv -4 \pmod {7}
Example 2.
p = 11, in this case p ≡ 3 (mod 4) and we have: (\tfrac{11}{5}) = +1: \qquad  \tfrac{1}{2}\left (5(\tfrac{11}{5})+3 \right )=4, \quad \tfrac{1}{2} \left (5(\tfrac{11}{5})- 3 \right )=1.
F_5=5 \text{ and } F_6=8.
5F_5^2=125\equiv 4 \pmod {11} \;\;\text{ and }\;\;5F_6^2=320\equiv 1 \pmod {11}
Example 3.
p = 13, in this case p ≡ 1 (mod 4) and we have: (\tfrac{13}{5}) = -1: \qquad \tfrac{1}{2}\left (5(\tfrac{13}{5})-5 \right ) =-5, \quad \tfrac{1}{2}\left (5(\tfrac{13}{5})+ 5 \right )=0.
F_6=8 \text{ and } F_7=13.
5F_6^2=320\equiv -5 \pmod {13} \;\;\text{ and }\;\;5F_7^2=845\equiv 0 \pmod {13}
Example 4.
p = 29, in this case p ≡ 1 (mod 4) and we have: (\tfrac{29}{5}) = +1: \qquad \tfrac{1}{2}\left (5(\tfrac{29}{5})-5 \right )=0, \quad \tfrac{1}{2}\left (5(\tfrac{29}{5})+5 \right )=5.
F_{14}=377 \text{ and } F_{15}=610.
5F_{14}^2=710645\equiv 0 \pmod {29} \;\;\text{ and }\;\;5F_{15}^2=1860500\equiv 5 \pmod {29}
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.
For example, F_1 = 1, F_3 = 2, F_5 = 5, F_7 = 13, F_9 = 34 = 2 \cdot 17, F_{11} = 89, F_{13} = 233, F_{15} = 610 = 2 \cdot 5 \cdot 61.
All known factors of Fibonacci numbers F(i) for all i < 50000 are collected at the relevant repositories.
collects all known factors of F(i) with i < 10000.
collects all known factors of F(i) with 10000 < i < 50000.
Periodicity modulo ''n''
If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n.
The lengths of the periods for various n form the so-called Pisano periods .
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.
Generalizations
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.
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.
Applications
Mathematics
The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):
The generating function can be expanded into \frac{x}{1-x-x^2} = x + x^2(1+x) + x^3(1+x)^2 + \dots + x^{k+1}(1+x)^k + \dots = \sum\limits_{n=0}^\infty F_nx^n and collecting like terms of x^n, we have the identity F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}.
To see how the formula is used, we can arrange the sums by the number of terms present:
{|
which is \binom{5}{0}+\binom{4}{1}+\binom{3}{2}, where we are choosing the positions of k twos from n-k-1 terms.
thumb|right|Use of the Fibonacci sequence to count  compositions These numbers also give the solution to certain enumerative problems, the most common of which is that of counting the number of ways of writing a given number  as an ordered sum of 1s and 2s (called compositions); there are  ways to do this (equivalently, it's also the number of domino tilings of the 2\times n rectangle).
For example, there are  ways one can climb a staircase of 5 steps, taking one or two steps at a time:
{|
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  without consecutive s is the Fibonacci number .
For example, out of the 16 binary strings of length 4, there are  without consecutive s – they are 0000, 0001,  0010, 0100, 0101, 1000, 1001, and 1010.
Equivalently,  is the number of subsets  of  without consecutive integers, that is, those  for which  for every .
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  without an odd number of consecutive s is the Fibonacci number .
For example, out of the 16 binary strings of length 4, there are  without an odd number of consecutive s – they are 0000, 0011, 0110, 1100, 1111.
Equivalently, the number of subsets  of  without an odd number of consecutive integers is .
A bijection with the sums to n is to replace 1 with 0 and 2 with 11.
The number of binary strings of length  without an even number of consecutive s or s is .
For example, out of the 16 binary strings of length 4, there are  without an even number of consecutive s or s – 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..
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 (F_n F_{n+3})^2 + (2 F_{n+1}F_{n+2})^2 = {F_{2 n+3}}^2.
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.
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.; see especially Lemma 8.2 (Ring Lemma), pp.
73–74, and Appendix B, The Ring Lemma, pp.
318–321.
Computer science
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.
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.
English translation by Myron J. Ricci in Soviet Mathematics - Doklady, 3:1259–1263, 1962.
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.
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.
They are also used in planning poker, which is a step in estimating in software development projects that use the Scrum methodology.
Nature
Fibonacci sequences appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone, and the family tree of honeybees.
Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.
Field daisies most often have petals in counts of Fibonacci numbers.
In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.
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.
thumb|Illustration of Vogel's model for  A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel in 1979.
This has the form
\theta = \frac{2\pi}{\varphi^2} n,\  r = c \sqrt{n}
where  is the index number of the floret and  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 , the nearest neighbors of floret number  are those at  for some index , which depends on , 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, typically counted by the outermost range of radii.
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, , is the number of female ancestors, which is , plus the number of male ancestors, which is .
This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
thumb|360px|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".)
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.
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 (F_1=1), and at his parents' generation, his X chromosome came from a single parent (F_2=1).
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 (F_3=2).
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 (F_4=3).
Five great-great-grandparents contributed to the male descendant's X chromosome (F_5=5), 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.
Other
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  reflections, for , is the kth Fibonacci number.
(However, when , there are three reflection paths, not two, one for each of the three surfaces.)
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.
The measured values of voltages and currents in the infinite resistor chain circuit (also called the resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence.
The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers.
The equivalent resistance of the entire circuit equals the golden ratio.
Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics.
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.
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.
See also .
See also
Elliott wave principle
Embree–Trefethen constant
The Fibonacci Association
Fibonacci numbers in popular culture
Fibonacci word
Strong law of small numbers
Verner Emil Hoggatt Jr.
Wythoff array
Fibonacci retracement
References
Footnotes
Citations Works cited
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External links
Periods of Fibonacci Sequences Mod m at MathPages
Scientists find clues to the formation of Fibonacci spirals in nature
