Show
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 Fn, 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:[1] 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...The Fibonacci numbers were first described in Indian mathematics,[2][3][4] 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.[5] 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 nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are 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[edit]The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; (see preceding image) The Fibonacci numbers may be defined by the recurrence relation[6] and for n > 1. Under some older definitions, the value is omitted, so that the sequence starts with and the recurrence is valid for n > 2.[7][8] The first 20 Fibonacci numbers Fn are:[1]
History[edit]Thirteen (F7) ways of arranging long and short syllables in a cadence of length six. Eight (F6) end with a short syllable and five (F5) end with a long syllable. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[3][9][10] In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1.[4] Knowledge of the Fibonacci sequence was expressed as early as Pingala (c. 450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for m beats (Fm+1) is obtained by adding one [S] to the Fm cases and one [L] to the Fm−1 cases.[11] Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).[12][2] 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):[10]
Hemachandra (c. 1150) is credited with knowledge of the sequence as well,[2] writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."[14][15] Outside India, the Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci[16][17] where it is used to calculate the growth of rabbit populations.[18][19] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?
At the end of the nth month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month n – 2) plus the number of pairs alive last month (month n – 1). The number in the nth month is the nth Fibonacci number.[20] In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At the end of the nth month, the number of pairs is equal to Fn. The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.[21] Relation to the golden ratio[edit]Closed-form expression[edit]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:[22] where is the golden ratio, and ψ is its conjugate:[23] Since , this formula can also be written as To see the relation between the sequence and these constants,[24] note that φ and ψ are both solutions of the equation so the powers of φ and ψ satisfy the Fibonacci recursion. In other words, and It follows that for any values a and b, the sequence defined by satisfies the same recurrence. If a and b are chosen so that U0 = 0 and U1 = 1 then the resulting sequence Un must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations: which has solution producing the required formula. Taking the starting values U0 and U1 to be arbitrary constants, a more general solution is: where Computation by rounding[edit]Since for all n ≥ 0, the number Fn is the closest integer to . Therefore, it can be found by rounding, using the nearest integer function: In fact, the rounding error is very small, being less than 0.1 for n ≥ 4, and less than 0.01 for n ≥ 8. Fibonacci numbers can also be computed by truncation, in terms of the floor function: As the floor function is monotonic, the latter formula can be inverted for finding the index n(F) of the smallest Fibonacci number that is not less than a positive integer F: where , , and Magnitude[edit]Since Fn is asymptotic to , the number of digits in Fn is asymptotic to . As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits. More generally, in the base b representation, the number of digits in Fn is asymptotic to Limit of consecutive quotients[edit]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 [25][26] This convergence holds regardless of the starting values and , unless . 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, , because the ratios between consecutive Fibonacci numbers approaches . 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[edit]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 ≥ 1: For , it is also the case that and it is also the case that These expressions are also true for n < 1 if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule Identification[edit]Binet's formula provides a proof that a positive integer x is a Fibonacci number if and only if at least one of or is a perfect square.[27] This is because Binet's formula, which can be written as , can be multiplied by and solved as a quadratic equation in via the quadratic formula: Comparing this to , it follows that In particular, the left-hand side is a perfect square. Matrix form[edit]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: For a given n, this matrix can be computed in O(log(n)) arithmetic operations, using the exponentiation by squaring method. Taking the determinant of both sides of this equation yields Cassini's identity, Moreover, since An Am = An+m for any square matrix A, the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing n into n + 1), In particular, with m = n, These last two identities provide a way to compute Fibonacci numbers recursively in O(log(n)) arithmetic operations and in time O(M(n) log(n)), where M(n) is the time for the multiplication of two numbers of n digits. This matches the time for computing the nth Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization).[28] Combinatorial identities[edit]Combinatorial proofs[edit]Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that can be interpreted as the number of [possibly empty] sequences of 1s and 2s whose sum is . This can be taken as the definition of with the conventions , meaning no such sequence exists whose sum is −1, and , meaning the empty sequence "adds up" to 0. In the following, is the cardinality of a set: In this manner the recurrence relation may be understood by dividing the sequences into two non-overlapping sets where all sequences either begin with 1 or 2: Excluding the first element, the remaining terms in each sequence sum to or and the cardinality of each set is or giving a total of sequences, showing this is equal to .
This may be seen by dividing all sequences summing to based on the location of the first 2. Specifically, each set consists of those sequences that start until the last two sets each with cardinality 1.
... where the last two terms have the value . From this it follows that . 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 is the (2n)th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to is the (2n + 1)th Fibonacci number minus 1.[30]
or in words, the sum of the squares of the first Fibonacci numbers up to is the product of the nth and (n + 1)th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size and decompose it into squares of size ; from this the identity follows by comparing areas: Symbolic method[edit]The sequence is also considered using the symbolic method.[31] More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is . Indeed, as stated above, the -th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of using terms 1 and 2. It follows that the ordinary generating function of the Fibonacci sequence, i.e. , is the complex function . Induction proofs[edit]Fibonacci identities often can be easily proved using mathematical induction. For example, reconsider Adding to both sides gives and so we have the formula for Similarly, add to both sides of to give Binet formula proofs[edit]The Binet formula is This can be used to prove Fibonacci identities. For example, to prove that note that the left hand side multiplied by becomes as required, using the facts and to simplify the equations. Other identities[edit]Numerous other identities can be derived using various methods. Here are some of them:[32] Cassini's and Catalan's identities[edit]Cassini's identity states that Catalan's identity is a generalization: d'Ocagne's identity[edit]
where Ln is the n'th Lucas number. The last is an identity for doubling n; other identities of this type are by Cassini's identity. These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number. More generally,[32] or alternatively Putting k = 2 in this formula, one gets again the formulas of the end of above section Matrix form. Generating function[edit]The generating function of the Fibonacci sequence is the power series This series is convergent for and its sum has a simple closed-form:[33] This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum: Solving the equation for s(x) results in the closed form. The partial fraction decomposition is given by where is the golden ratio and is its conjugate. gives the generating function for the negafibonacci numbers, and satisfies the functional equation Reciprocal sums[edit]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 and the sum of squared reciprocal Fibonacci numbers as If we add 1 to each Fibonacci number in the first sum, there is also the closed form and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio, The sum of all even-indexed reciprocal Fibonacci numbers is[34] with the Lambert series since So the reciprocal Fibonacci constant is Moreover, this number has been proved irrational by Richard André-Jeannin.[35] The Millin series gives the identity[36]
which follows from the closed form for its partial sums as N tends to infinity: Primes and divisibility[edit]Divisibility properties[edit]Every third number of the sequence is even (a multiple of ) and, more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property[37][38] where gcd is the greatest common divisor function. In particular, any three consecutive Fibonacci numbers are pairwise coprime because both and . That is, for every n. Every prime number p divides a Fibonacci number that can be determined by the value of p modulo 5. If p is congruent to 1 or 4 (mod 5), then p divides Fp − 1, and if p is congruent to 2 or 3 (mod 5), then, p divides Fp + 1. The remaining case is that p = 5, and in this case p divides Fp. These cases can be combined into a single, non-piecewise formula, using the Legendre symbol:[39] Primality testing[edit]The above formula can be used as a primality test in the sense that if where the Legendre symbol has been replaced by the Jacobi symbol, then this is evidence that n is a prime, and if it fails to hold, then n is definitely not a prime. If n is composite and satisfies the formula, then n is a Fibonacci pseudoprime. When m is large – say a 500-bit number – then we can calculate Fm (mod n) efficiently using the matrix form. Thus Here the matrix power Am is calculated using modular exponentiation, which can be adapted to matrices.[40] Fibonacci primes[edit]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.[41] Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers. No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number.[42] The only nontrivial square Fibonacci number is 144.[43] Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.[44] In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.[45] 1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.[46] No Fibonacci number can be a perfect number.[47] More generally, no Fibonacci number other than 1 can be multiply perfect,[48] and no ratio of two Fibonacci numbers can be perfect.[49] Prime divisors[edit]With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).[50] As a result, 8 and 144 (F6 and F12) are the only Fibonacci numbers that are the product of other Fibonacci numbers 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 [51][52] For example, It is not known whether there exists a prime p such that Such primes (if there are any) would be called Wall–Sun–Sun primes. Also, if p ≠ 5 is an odd prime number then:[53] Example 1. p = 7, in this case p ≡ 3 (mod 4) and we have: Example 2. p = 11, in this case p ≡ 3 (mod 4) and we have: Example 3. p = 13, in this case p ≡ 1 (mod 4) and we have:
Example 4. p = 29, in this case p ≡ 1 (mod 4) and we have: For odd n, all odd prime divisors of Fn are congruent to 1 modulo 4, implying that all odd divisors of Fn (as the products of odd prime divisors) are congruent to 1 modulo 4.[54] For example, All known factors of Fibonacci numbers F(i) for all i < 50000 are collected at the relevant repositories.[55][56] Periodicity modulo n[edit]If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n.[57] The lengths of the periods for various n form the so-called Pisano periods 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. Generalizations[edit]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:
Applications[edit]Mathematics[edit]The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of Pascal's triangle. The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):[59] The generating function can be expanded into and collecting like terms of , we have the identity To see how the formula is used, we can arrange the sums by the number of terms present:
which is , where we are choosing the positions of k twos from n-k-1 terms. Use of the Fibonacci sequence to count {1, 2}-restricted compositions These numbers also give the solution to certain enumerative problems,[60] the most common of which is that of counting the number of ways of writing a given number n as an ordered sum of 1s and 2s (called compositions); there are Fn+1 ways to do this (equivalently, it's also the number of domino tilings of the rectangle). For example, there are F5+1 = F6 = 8 ways one can climb a staircase of 5 steps, taking one or two steps at a time:
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.
Computer science[edit]Fibonacci tree of height 6. Balance factors green; heights red.
Nature[edit]Yellow chamomile head showing the arrangement in 21 (blue) and 13 (aqua) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants. Fibonacci sequences appear in biological settings,[70] such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[71] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone,[72] and the family tree of honeybees.[73][74] Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.[75] Field daisies most often have petals in counts of Fibonacci numbers.[76] In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.[77] Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.[78] Illustration of Vogel's model for n = 1 ... 500 A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel [de] in 1979.[79] This has the form where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j, which depends on r, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,[80] typically counted by the outermost range of radii.[81] Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules:
Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, Fn, is the number of female ancestors, which is Fn−1, plus the number of male ancestors, which is Fn−2.[82] This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated. The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships".[83]) It has been noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence.[83] A male individual has an X chromosome, which he received from his mother, and a Y chromosome, which he received from his father. The male counts as the "origin" of his own X chromosome (), and at his parents' generation, his X chromosome came from a single parent (). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome (). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome (). Five great-great-grandparents contributed to the male descendant's X chromosome (), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a population founder appears on all lines of the genealogy.) Other[edit]
See also[edit]
References[edit]Footnotes
Citations
Works cited[edit]
External links[edit]
|