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

**Mathematics** includes the study of such topics as quantity, structure, space, and change. It has no generally accepted definition.

In mathematics, an **integer sequence** is a sequence of integers.

- History
- Applications
- Music
- Nature
- Mathematics
- Sequence properties
- Relation to the golden ratio
- Matrix form
- Identification
- Combinatorial identities
- Other identities
- Power series
- Reciprocal sums
- Primes and divisibility
- Right triangles
- Magnitude
- Generalizations
- See also
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and

for *n* > 1.

One has *F*_{2} = 1. In some books, and particularly in old ones, *F*_{0}, the "0" is omitted, and the Fibonacci sequence starts with *F*_{1} = *F*_{2} = 1.^{ [2] }^{ [3] } The beginning of the sequence is thus:

^{ [4] }

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.

In mathematics, two quantities are in the **golden ratio** if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities *a* and *b* with *a* > *b* > 0,

Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. They appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. 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] }

**Fibonacci** was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, "Fibonacci", was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for *filius Bonacci*. He is also known as **Leonardo Bonacci**, **Leonardo of Pisa**, or **Leonardo Bigollo** ("traveller") **Pisano**.

Acharya **Pingala** was an ancient Indian mathematician who authored the *Chandaḥśāstra*, the earliest known treatise on Sanskrit prosody.

* Liber Abaci* is a 1202 historic book on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci.

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.

The * Fibonacci Quarterly* is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding editors were Verner Emil Hoggatt, Jr. and Alfred Brousseau; the present editor is Professor Curtis Cooper of the Mathematics Department of the University of Central Missouri.

In computer science, the **Fibonacci search technique** is a method of searching a sorted array using a divide and conquer algorithm that narrows down possible locations with the aid of Fibonacci numbers. Compared to binary search where the sorted array is divided into two equal-sized parts, one of which is examined further, Fibonacci search divides the array into two parts that have sizes that are consecutive Fibonacci numbers. On average, this leads to about 4% more comparisons to be executed, but it has the advantage that one only needs addition and subtraction to calculate the indices of the accessed array elements, while classical binary search needs bit-shift, division or multiplication, operations that were less common at the time Fibonacci search was first published. Fibonacci search has an average- and worst-case complexity of *O*(log *n*).

In computer science, a **Fibonacci heap** is a data structure for priority queue operations, consisting of a collection of heap-ordered trees. It has a better amortized running time than many other priority queue data structures including the binary heap and binomial heap. Michael L. Fredman and Robert E. Tarjan developed Fibonacci heaps in 1984 and published them in a scientific journal in 1987. Fibonacci heaps are named after the Fibonacci numbers, which are used in their running time analysis.

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.

In botany, **phyllotaxis** or **phyllotaxy** is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature.

The **pineapple** is a tropical plant with an edible fruit, also called pineapples, and the most economically significant plant in the family Bromeliaceae.

The **globe artichoke**, also known as **French artichoke** and **green artichoke** in the USA, is a variety of a species of thistle cultivated as a food.

Fibonacci numbers are also closely related to Lucas numbers in that they form a complementary pair of Lucas sequences and . Lucas numbers are also intimately connected with the golden ratio.

The Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985.^{ [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 * (1202) by Fibonacci.^{ [6] }^{ [17] } using it to calculate the growth of rabbit populations.^{ [18] }^{ [19] } Fibonacci considers the growth of a hypothetical, idealized (biologically unrealistic) rabbit population, assuming that: a newly born 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. 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 the female produces a new pair, so now there are 2 pairs of rabbits in the field.
- At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
- At the end of the fourth month, the original female has produced yet another new pair, and the female born two months ago also produces her 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 new pairs (that is, the number of pairs in month *n* − 2) plus the number of pairs alive last month (that is, *n* − 1). This is the *n*th Fibonacci number.^{ [20] }

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

- The Fibonacci numbers are important in the 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.
^{ [22] } - Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics.
^{ [23] }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. - 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.
^{ [24] } - 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. - Fibonacci numbers are used by some pseudorandom number generators.
- They are also used in planning poker, which is a step in estimating in software development projects that use the Scrum methodology.
- 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*. - 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 one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers.
^{ [25] } - 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.
^{ [26] }^{ [27] } - 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.^{ [28] } - 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.)^{ [29] } - Mario Merz included the Fibonacci sequence in some of his works beginning in 1970.
^{ [30] } - Fibonacci retracement levels are widely used in technical analysis for financial market trading.

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.^{ [31] }

Fibonacci sequences appear in biological settings,^{ [32] } such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,^{ [33] } the flowering of artichoke, an uncurling fern and the arrangement of a pine cone,^{ [34] } and the family tree of honeybees.^{ [35] }^{ [36] } Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.^{ [37] } Field daisies most often have petals in counts of Fibonacci numbers.^{ [38] } In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.^{ [39] }

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.^{ [40] }

A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel in 1979.^{ [41] } 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,^{ [42] } typically counted by the outermost range of radii.^{ [43] }

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}.^{ [44] } This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

Luke Hutchison noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence.^{ [45] } 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. (Note that 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.^{ [46] }

The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):^{ [47] }

These numbers also give the solution to certain enumerative problems.^{ [48] } The most common is that of counting the number of compositions of 1s and 2s which sum to a given total *n*: there are *F*_{n+1} ways to do this.

For example, if *n* = 5, then *F*_{n+1} = *F*_{6} = 8 counts the eight compositions summing to 5:

1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2.

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. By symmetry, the number of strings of length*n*without consecutive 0s is also*F*_{n+2}. Equivalently,*F*_{n+2}is the number of subsets S ⊂ {1,...,n} without consecutive integers: {i, i+1} ⊄ S for every i. The symmetric statement is:*F*_{n+2}is the number of subsets S ⊂ {1,...,n} without two consecutive skipped integers: that is, S = {a_{1}< ... < a_{k}} with a_{i+1}≤ a_{i}+ 2. - 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 ⊂ {1,...,n} without an odd number of consecutive integers is*F*_{n+1}. - 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.

The first 21 Fibonacci numbers *F _{n}* for

*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^{ [50] } 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 solution. It has become known as "Binet's formula", though it was already known by Abraham de Moivre and Daniel Bernoulli:^{ [51] }

where

is the golden ratio ( OEIS: A001622 ), and

^{ [52] }

Since , this formula can also be written as

To see this,^{ [53] } 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, that is by the use of the nearest integer function:

or in terms of the floor function:

Similarly, if we already know that the number *F* > 1 is a Fibonacci number, we can determine its index within the sequence by

where can be computed using logarithms to other usual bases. For example, .

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 ^{ [54] }^{ [55] }

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).^{ [56] }

The question may arise whether a positive integer *x* is a Fibonacci number. This is true if and only if one or both of or is a perfect square.^{ [57] } 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.^{ [58] } 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.^{ [59] }

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 note that 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.

The sequence is also considered using the symbolic method.^{ [60] } More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is . Indeed, as stated above, the -th Fibonacci numbes equals the number of way to partition using segments of size 1 or 2.

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

Numerous other identities can be derived using various methods. Some of the most noteworthy are:^{ [61] }

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,^{ [61] }

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:^{ [62] }

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 above closed form.

Setting *x* = 1/*k*, the closed form of the series becomes

In particular, if k is an integer greater than 1, then this series converges. Further setting *k* = 10^{m} yields

for all positive integers m.

Some math puzzle-books present as curious the particular value that comes from *m* = 1, which is ^{ [63] } Similarly, *m* = 2 gives

Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as

and the sum of squared reciprocal Fibonacci numbers as

If we add 1 to each Fibonacci number in the first sum, there is also the closed form

and there is a *nested* sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,

No closed formula for the reciprocal Fibonacci constant

is known, but the number has been proved irrational by Richard André-Jeannin.^{ [64] }

The **Millin series** gives the identity^{ [65] }

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 formula, using the Legendre symbol:^{ [68] }

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.^{ [69] }

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.^{ [70] }

*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.^{ [71] }

The only nontrivial square Fibonacci number is 144.^{ [72] } Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.^{ [73] } In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.^{ [74] }

1, 3, 21, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.^{ [75] }

No Fibonacci number can be a perfect number.^{ [76] } More generally, no Fibonaci number other than 1 can be multiply perfect,^{ [77] } and no ratio of two Fibonacci numbers can be perfect.^{ [78] }

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).^{ [79] } 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

^{ [80] }^{ [81] }

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:^{ [82] }

**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.^{ [83] }

For example,

All known factors of Fibonacci numbers *F*(*i*) for all *i* < 50000 are collected at the relevant repositories.^{ [84] }^{ [85] }

If the members of the Fibonacci sequence are taken mod *n*, the resulting sequence is periodic with period at most *6n*.^{ [86] } 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.

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:^{ [87] }

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.
^{ [61] } - 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}. - 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*.^{ [88] } - Adding other objects than integers, for example functions or strings – one essential example is Fibonacci polynomials.

A **complex number** is a number that can be expressed in the form *a* + *bi*, where *a* and *b* are real numbers, and *i* is a solution of the equation *x*^{2} = −1. Because no real number satisfies this equation, *i* is called an imaginary number. For the complex number *a* + *bi*, *a* is called the **real part**, and *b* is called the **imaginary part**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

In number theory, the **Legendre symbol** is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number *p*: its value at a (nonzero) quadratic residue mod *p* is 1 and at a non-quadratic residue (*non-residue*) is −1. Its value at zero is 0.

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 *φ*(*n*) or *ϕ*(*n*), 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.

**Fermat's little theorem** states that if p is a prime number, then for any integer a, the number *a*^{p} − *a* is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

In mathematics, the **special unitary group** of degree *n*, denoted SU(*n*), is the Lie group of *n* × *n* unitary matrices with determinant 1.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

In number theory, a **Wieferich prime** is a prime number *p* such that *p*^{2} divides 2^{p − 1} − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime *p* divides 2^{p − 1} − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem, at which time both of Fermat's theorems were already well known to mathematicians.

In number theory, a **Wall–Sun–Sun prime** or **Fibonacci–Wieferich prime** is a certain kind of prime number which is conjectured to exist, although none are known.

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.

In mathematics, the **random Fibonacci sequence** is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation *f*_{n} = *f*_{n−1} ± *f*_{n−2}, where the signs + or − are chosen at random with equal probability 1/2, independently for different *n*. By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943…(sequence A078416 in the OEIS), a mathematical constant that was later named **Viswanath's constant**.

In mathematics, an **asymptotic expansion**, **asymptotic series** or **Poincaré expansion** is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by Dingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

In number theory, the *n*th **Pisano period**, written π(*n*), is the period with which the sequence of Fibonacci numbers taken modulo *n* repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774.

In number theory, the **Carmichael function** associates to every positive integer *n* a positive integer , defined as the smallest positive integer *m* such that

A **Fibonacci word** is a specific sequence of binary digits. The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.

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

The following **examples of generating functions** are in the spirit of George Pólya, who advocated learning mathematics by doing and re-capitulating as many examples and proofs as possible. The purpose of this article is to present common ways of creating generating functions.

**Feller's coin-tossing constants** are a set of numerical constants which describe asymptotic probabilities that in *n* independent tosses of a fair coin, no run of *k* consecutive heads appears.

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, **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.

**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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*Exploring the World of Mathematics: From Ancient Record Keeping to the Latest Advances in Computers*. New Leaf Publishing Group. ISBN 978-1-61458-155-0. - 1 2 Pisano 2002, pp. 404–5.
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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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*Mathematical Circus*, The Mathematical Association of America, p. 153, ISBN 978-0-88385-506-5,It is ironic that Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Édouard Lucas... attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber abaci

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