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In mathematics, the Fibonacci numbers form a sequence defined recursively by: F(n):= egin 0 & mbox n = 0; \ 1 & mbox n = 1; \ F(n-1)+F(n-2) & mbox n > 1. \ end That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers , also denoted as Fn, for n = 0, 1, … , are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711. The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India. Origins The Fibonacci numbers first appear, under the name maatraameru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chhandah-shāstra, the Art of Prosody, 450 or 200 BC). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well known text on these. A commentary on Virahanka by Gopala in the 12th c. also revisits the problem in some detail. Sanskrit vowel sounds can be long (L) or short (S), and Virahanka's analysis, which came to be known as mAtrA-vritta wishes to compute how many metres (mAtrAs) of a given overall length can be composed of these syllables. If the long syllable is twice as long as the short, the solutions are: 1 mora: S (1 pattern) 2 morae: SS; L (2) 3 morae: SSS, SL; LS (3) 4 morae: SSSS, SSL, SLS; LSS, LL (5) 5 morae: SSSSS, SSSL, SSLS, SLSS, SLL; LSSS, LSL, LLS (8) A pattern of length n can be formed by adding S to a pattern of length n−1, or L to a pattern of length n−2; and the prosodicists showed that the number of patterns of length n is the sum of the two previous numbers in the series. Donald Knuth reviews this work in The Art of Computer Programming as equivalent formulations of the bin packing problem for items of lengths 1 and 2. In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci (1202). He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that: Let the population at month n be F(n). At this time, only rabbits who were alive at month n−2 are fertile and produce offspring, so F(n−2) pairs are added to the current population of F(n−1). Thus the total is F(n) = F(n−1) + F(n−2).• The bee ancestry code Fibonacci numbers also appear in the description of the reproduction of a population of idealized bees, according to the following rules: If one traces the ancestry of this male bee (1 bee), he has 1 female parent (1 bee). This female had 2 parents, a male and a female (2 bees). The female had two parents, a male and a female, and the male had one female (3 bees). Those two females each had two parents, and the male had one (5 bees). If one continues this sequence, it gives a perfectly accurate depiction of the Fibonacci sequence. Notice that this is a mathematical statement, it does not describe actual bee ancestries. In reality, some ancestors of a particular bee will always be sisters or brothers, thus breaking the lineage of distinct parents. Relation to the golden ratio The golden ratio (phi), is defined as the ratio that results when a line is divided so that the whole line has the same ratio to the larger segment as the larger segment has to the smaller segment. Expressed mathematically, normalising the larger part to unit length, it is the positive solution of the equation: or equivalently which is equal to Closed form expression Like every sequence defined by linear recursion, the Fibonacci numbers have a closed-form solution. It has become known as Binet's formula: ight) = , , where is the golden ratio defined above. Note the similarity of the Fibonacci recursion to the defining equation of the golden ratio in the form also known as the generating polynomial of the recursion. Proof (by induction): Any root of the equation above satifies and multiplying by shows: By definition is a root of the equation, and the other root is . Therefore: and Now consider the functions: defined for any real All these functions satisfy the Fibonacci recursion Selecting and gives the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore: and establishing the base cases of the induction, proving that for all For any two starting values, a combination can be found such that the function is the exact closed formula for the series. Since for all is the closest integer to For computational purposes, this is expressed using the floor function: floor. Limit of consecutive quotients Johannes Kepler pointed out that the ratio of consecutive Fibonacci numbers converges to the golden ratio as the limit. This convergence does not depend on the starting values chosen, excluding 0, 0. Proof: It follows from the explicit formula that for any real : because and thus Matrix form A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is or The eigenvalues of the matrix A are and , and the elements of the eigenvectors of A, and , are in the ratios and . Note that this matrix has a determinant of −1, and thus it is a 2×2 unimodular matrix. This property can be understood in terms of the continued fraction representation for the golden mean: = 1; 1, 1, 1, 1, …. The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed expression for the Fibonacci numbers: egin F_ & F_n \ F_n & F_ end. Taking the determinant of both sides of this equation yields the identity Additionally, since for any square matrix , the following identities can be derived: Applications The Fibonacci numbers are important in the 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. Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem. The Fibonacci numbers occur in the sums of diagonals in Pascal's triangle and Lozanić's triangle (see "Binomial coefficient"). 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. Fibonacci numbers are used by some pseudorandom number generators. A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers •. In music Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements. Examples include Béla Bartók's Music for Strings, Percussion, and Celesta. Since the conversion factor 1.609 for miles to kilometers is close to the golden ratio (denoted φ), the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead. Fibonacci numbers in nature
Identities F(n + 1) = F(n) + F(n − 1) F(0) + F(1) + F(2) + … + F(n) = F(n + 2) − 1 F(1) + 2 F(2) + 3 F(3) + … + n F(n) = n F(n + 2) − F(n + 3) + 2 F(0)2 + F(1)2 + F(2)2 + … + F(n)2 = F(n) F(n + 1) These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using combinatorial arguments here. In particular, F(n) can be interpreted as the number of ways summing 1's and 2's to n − 1, with the convention that F(0) = 0, meaning no sum will add up to −1, and that F(1) = 1, meaning the empty sum will "add up" to 0. Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice. Proof of the first identity. Without loss of generality, we may assume n ≥ 1. Then F(n + 1) counts the number of ways summing 1's and 2's to n. When the first summand is 1, there are F(n) ways to complete the counting for n − 1; and the first summand is 2, there are F(n − 1) ways to complete the counting for n − 2. Thus, in total, there are F(n) + F(n − 1) ways to complete the counting for n. Proof of the second identity. We count the number of ways summing 1's and 2's to n + 1 such that at least one of the summands is 2. As before, there are F(n + 2) ways summing 1's and 2's to n + 1 when n ≥ 0. Since there is only one sum of n + 1 that does not use any 2, namely 1 + … + 1 (n + 1 terms), we subtract 1 from F(n + 2). Equivalently, we can consider the first occurrence of 2 as a summand. If, in a sum, the first summand is 2, then there are F(n) ways to the complete the counting for n − 1. If the second summand is 2 but the first is 1, then there are F(n − 1) ways to complete the counting for n − 2. Proceed in this fashion. Eventually we consider the (n + 1)th summand. If it is 2 but all of the previous n summands are 1's, then there are F(0) ways to complete the counting for 0. If a sum contains 2 as a summand, the first occurrence of such summand must take place in between the first and (n + 1)th position. Thus F(n) + F(n − 1) + … + F(0) gives the desired counting. Proof of the third identity. This identity can be established in two stages. First, we count the number of ways summing 1s and 2s to −1, 0, …, or n + 1 such that at least one of the summands is 2. By our second identity, there are F(n + 2) − 1 ways summing to n + 1; F(n + 1) − 1 ways summing to n; …; and, eventually, F(2) − 1 way summing to 1. As F(1) − 1 = F(0) = 0, we can add up all n + 1 sums and apply the second identity again to obtain ''F''(''n'' + 2) − 1 + ''F''(''n'' + 1) − 1 + … + ''F''(2) − 1 = ''F''(''n'' + 2) − 1 + ''F''(''n'' + 1) − 1 + … + ''F''(2) − 1 + ''F''(1) − 1 + F(0) = F(n + 2) + ''F''(''n'' + 1) + … + ''F''(1) + ''F''(0) − (n + 2) = F(n + 2) + F(n + 3) − (n + 2). On the other hand, we observe from the second identity that there are Since the two methods of counting refer to the same number, we have (n + 1) F(0) + n F(1) + … + F(n) = F(n + 2) + F(n + 3) − (n + 2) Finally, we complete the proof by subtracting the above identity from n + 1 times the second identity. Common factors Any two consecutive Fibonacci numbers are relatively prime. Suppose that Fn and Fn+1 have a common factor g. Then Fn−1 = Fn+1 – Fn must also be a multiple of g; and by induction the same must be true of all lower Fibonacci numbers. But F1 = 1, so g = 1. Other identities include relationships to the Lucas numbers, which have the same recursive properties but start with L0=2 and L1=1. These properties include F2n=FnLn Identity for doubling n Another identity useful for calculating Fn for large values of n is for all integers n and k. Power series The Fibonacci power series has a simple and interesting closed-form solution for x < 1/φ: This function is therefore the generating function of the Fibonacci sequence. It can be proven as follows: Substituting : Therefore, In particular, math puzzle-books note the curious value . The sum is easily proved by noting that and then explicitly evaluating the sum. Reciprocal sums Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. The reciprocal Fibonacci constant has been proved irrational by Richard André-Jeannin, but no closed form expression for it is known. Extension to negative integers Using Fn-2 = Fn - Fn-1, one can extend the Fibonacci numbers to negative integers. So we get: ... -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, ... and F-n = -(-1)nFn. Vector space The term Fibonacci sequence is also applied more generally to any function g from the integers to a field for which g(n+2) = g(n) + g(n+1). These functions are precisely those of the form g(n) = F(n)g(1) + F(n-1)g(0), so the Fibonacci sequences form a vector space with the functions F(n) and F(n-1) as a basis. More generally, the range of g may be taken to be any abelian group (regarded as a Z-module). Then the Fibonacci sequences form a 2-dimensional Z-module in the same way. Lucas numbers In particular, the Fibonacci sequence L with L(1) = 1 and L(2) = 3 is referred to as the Lucas numbers, after Edouard Lucas. This sequence was described by Leonhard Euler in 1748, in the Introductio in Analysin Infinitorum. The significance in the Lucas numbers L(n) lies in the fact that raising the golden ratio to the nth power yields ight) ight)^n = rac 1 2 left( L(n) + F(n) sqrt ight). Lucas numbers are related to Fibonacci numbers by the relation ight)=Fleft(n-1 ight)+Fleft(n+1 ight)., A generalization of the Fibonacci sequence are the Lucas sequences. One kind can be defined thus: U(0) = 0 U(1) = 1 U(n + 2) = PU(n + 1) − QU(n) where the normal Fibonacci sequence is the special case of P = 1 and Q = −1. Another kind of Lucas sequence begins with V(0) = 2, V(1) = P. Such sequences have applications in number theory and primality proving. The Padovan sequence is generated by the recurrence P(n) = P(n − 2) + P(n − 3). Tribonacci numbers The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are : 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, … The tribonacci constant is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x3 − x2 − x − 1, approximately 1.83929, and also satisfies the equation x + x−3 = 2. It is important in the study of the snub cube. The tribonacci numbers are also given by where the outer brackets denote the nearest integer function and ight)^ ight)^ (Simon Plouffe, 1993).* Tetranacci numbers The tetranacci numbers start with four predetermined terms, each term afterwards being the sum of the preceding four terms. The first few tetranacci numbers are : 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, … The tetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is a root of the polynomial x4 − x3 − x2 − x − 1, approximately 1.92756, and also satisfies the equation x + x−4 = 2. Other -anacci numbers Pentanacci, hexanacci and heptanacci numbers have been computed, with perhaps less interest so far in research. Interestingly, there is a limit to this with increasing n. A 'polynacci' sequence, if one could be described, would after an infinite number of zeroes yield the sequence ..., 0, 0, 1, 1, 2, 4, 8, 16, 32, ... which are simply powers of 2. Other generalizations The Fibonacci polynomials are another generalization of Fibonacci numbers. A random Fibonacci sequence can be defined by tossing a coin for each position n of the sequence and taking F(n)=F(n−1)+F(n−2) if it lands heads and F(n)=F(n−1)−F(n−2) if it lands tails. Work by Furstenburg and Kesten guarantees that this sequence almost surely grows exponentially at a constant rate: the constant is independent of the coin tosses and was computed in 1999 by Divakar Viswanath. It is now known as Viswanath's constant. A repfigit or Keith number is an integer, that when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4,7,11,18,29,47) reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the number, a tetranacci number if the number has four digits, etc. The first few repfigits are : 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, … Since the set of sequences satisfying the relation S(n) = S(n−1) + S(n−2) is closed under termwise addition and under termwise multiplication by a constant, it can be viewed as a vector space. Any such sequence is uniquely determined by a choice of two elements, so the vector space is two-dimensional. If we abbreviate such a sequence as (S(0), S(1)), the Fibonacci sequence F(n) = (0, 1) and the shifted Fibonacci sequence F(n−1) = (1, 0) are seen to form a canonical basis for this space, yielding the identity: S(n) = S(0)F(n−1) + S(1)F(n) for all such sequences S. For example, if S is the Lucas sequence 1, 3, 4, 7, 11…, then we obtain L(n) = F(n−1) + 3F(n). Fibonacci primes The first few Fibonacci numbers that are also prime numbers are : 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …. It is not known if there are infinitely many Fibonacci primes. Fibonacci strings In analogy to its numerical counterpart, a Fibonacci string is defined by: F_n = F(n):= egin b & mbox n = 0; \ a & mbox n = 1; \ F(n-1)+F(n-2) & mbox n > 1. \ end , where + denotes the concatenation of two strings. The sequence of Fibonacci strings starts: b, a, ab, aba, abaab, abaababa, abaababaabaab, … The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number. Fibonacci strings appear as inputs for the worst case in some computer algorithms. Architecture Cinema Literature Music Television Visual Arts Games See also | |||||||||
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