From Wikipedia, the free encyclopedia
By definition, the first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s.
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation
with seed values
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonaccio, "son of Bonaccio"). Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.
The Fibonacci sequence was well known in ancient India, where it was applied to the metrical sciences (prosody), long before it was known in Europe. Developments have been attributed to Pingala (200 BC), Virahanka (6th century AD), Gopāla (c.1135 AD), and Hemachandra (c.1150 AD).
The motivation came from Sanskrit prosody, where long syllables have length 2 and short syllables have length 1. Any pattern of length n can be formed by adding a short syllable to a pattern of length n − 1, or a long syllable to a pattern of length n − 2; thus the prosodists showed that the number of patterns of length n is the sum of the two previous numbers in the sequence. Donald Knuth reviews this work in The Art of Computer Programming.
In the West, the sequence was studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202). He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that:
- In the "zero" month, there is one pair of rabbits (additional pairs of rabbits = 0).
- In the first month, the first pair begets another pair (additional pairs of rabbits = 1).
- In the second month, both pairs of rabbits have another pair, and the first pair dies (additional pairs of rabbits = 1).
- In the third month, the second pair and the new two pairs have a total of three new pairs, and the older second pair dies (additional pairs of rabbits = 2).
The laws of this are that each pair of rabbits has 2 pairs in its lifetime, and dies.
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).
 List of Fibonacci numbers
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765
Using the recurrence relation, the sequence can also be extended to negative index n. The result satisfies the equation
Thus the complete sequence is
 Divisibility properties
Every 3rd 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
 Relation to the golden ratio
 Closed form expression
(note, that , as can be seen from the defining equation above).
The Fibonacci recursion
is similar to the defining equation of the golden ratio in the form
which is also known as the generating polynomial of the recursion.
 Proof by induction
Any root of the equation above satisfies and multiplying by shows:
By definition is a root of the equation, and the other root is Therefore:
Both and are geometric series (for n = 1, 2, 3, ...) that satisfy the Fibonacci recursion. The first series grows exponentially; the second exponentially tends to zero, with alternating signs. Because the Fibonacci recursion is linear, any linear combination of these two series will also satisfy the recursion. These linear combinations form a two-dimensional linear vector space; the original Fibonacci sequence can be found in this space.
Linear combinations of series and , with coefficients a and b, can be defined by
- for any real
All thus-defined series satisfy the Fibonacci recursion
Requiring that Fa,b(0) = 0 and Fa,b(1) = 1 yields and , resulting in the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore, an explicit check can be made:
establishing the base cases of the induction, proving that
- for all
Therefore, for any two starting values, a combination a,b can be found such that the function is the exact closed formula for the series.
 Computation by rounding
 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 the limit approaches the golden ratio .
This convergence does not depend on the starting values chosen, excluding 0, 0. For example, the initial values 19 and 31 generate the sequence 19, 31, 50, 81, 131, 212, 343, 555 ... etc. The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
In brief, Fibonacci numbers are approximately exponential – where the constant depends on starting values – as the remaining term in the exact formula for the Fibonacci numbers becomes exponentially close to zero as n grows. Taking the ratio yields
More formally, it must always follow from the explicit formula that for any real
because and thus
 Decomposition of powers of 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 expression is also true for if the Fibonacci sequence is extended to negative integers using the Fibonacci rule
 Matrix form
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
The eigenvalues of the matrix A are and , and the elements of the eigenvectors of A, and , are in the ratios and Using these facts, and the properties of eigenvalues, we can derive a direct formula for the nth element in the fibonacci series:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The matrix representation gives the following closed expression for the Fibonacci numbers:
Taking the determinant of both sides of this equation yields Cassini's identity
Additionally, since AnAm = Am + n for any square matrix A, the following identities can be derived:
In particular, with m = n,
 Recognizing Fibonacci numbers
The question may arise whether a positive integer z is a Fibonacci number. Since F(n) is the closest integer to , the most straightforward, brute-force test is the identity
which is true if and only if z is a Fibonacci number. In this formula, F(n) can be computed rapidly using any of the previously discussed closed-form expressions.
(with coprime positive integers p, q) is true if and only if p and q are successive Fibonacci numbers. From this one derives the criterion that z is a Fibonacci number if and only if the closed interval
contains a positive integer.
Most identities involving Fibonacci numbers draw from combinatorial arguments. F(n) can be interpreted as the number of sequences of 1s and 2s that sum to n − 1, with the convention that F(0) = 0, meaning no sum will add up to −1, and that F(1) = 1, meaning the empty sum will "add up" to 0. Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice. This is discussed in further detail at Young–Fibonacci lattice.
 First identity
- The nth Fibonacci number is the sum of the previous two Fibonacci numbers.
We must establish that the sequence of numbers defined by the combinatorial interpretation above satisfy the same recurrence relation as the Fibonacci numbers (and so are indeed identical to the Fibonacci numbers).
The set of F(n + 1) ways of making ordered sums of 1s and 2s that sum to n may be divided into two non-overlapping sets. The first set contains those sums whose first summand is 1; the remainder sums to n − 1, so there are F(n) sums in the first set. The second set contains those sums whose first summand is 2; the remainder sums to n − 2, so there are F(n − 1) sums in the second set. The first summand can only be 1 or 2, so these two sets exhaust the original set. Thus F(n + 1) = F(n) + F(n−1).
 Second identity
- The sum of the first n Fibonacci numbers is the (n + 2)nd Fibonacci number minus 1.
We count the number of ways summing 1s and 2s to n + 1 such that at least one of the summands is 2.
As before, there are F(n + 2) ways summing 1s and 2s to n + 1 when n ≥ 0. Since there is only one sum of n + 1 that does not use any 2, namely 1 + ... + 1 (n + 1 terms), we subtract 1 from F(n + 2).
Equivalently, we can consider the first occurrence of 2 as a summand. If, in a sum, the first summand is 2, then there are F(n) ways to the complete the counting for n − 1. If the second summand is 2 but the first is 1, then there are F(n − 1) ways to complete the counting for n − 2. Proceed in this fashion. Eventually we consider the (n + 1)th summand. If it is 2 but all of the previous n summands are 1s, then there are F(0) ways to complete the counting for 0. If a sum contains 2 as a summand, the first occurrence of such summand must take place in between the first and (n + 1)th position. Thus F(n) + F(n − 1) + ... + F(0) gives the desired counting.
- For n = 0, , so the equation is true for n = 0.
- For n = x, assume .
- Add the next Fibonacci number Fx + 1 to both sides: .
- By the Fibonacci recurrence relation, Fx + 1 + Fx + 2 = Fx + 3, so , which is the n = x + 1 case, proving that where the equation is true for n = x, so is it for n = x + 1.
 Third identity
This identity has slightly different forms for Fj, depending on whether j is odd or even.
The sum of the first n − 1 Fibonacci numbers, Fj, such that j is odd, is the (2n)th Fibonacci number.
The sum of the first n Fibonacci numbers, Fj, such that j is even, is the (2n + 1)th Fibonacci number minus 1.
 1: j is odd
By induction for F2n:
A basis case for this could be F1 = F2.
 2: j is even
By induction for F2n+1:
A basis case for this could be F0 = F1 − 1.
 Alternative proof
By using identity 1 we can construct a telescoping sum:
If the summands are the Fibonacci numbers with even index, the proof is very similar. Summing both cases yields identity 2.
 Fourth 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) − 1] − (n + 2)
- = F(n + 2) + F(n + 3) − (n + 3).
On the other hand, we observe from the second identity that there are
- F(0) + F(1) + ... + F(n − 1) + F(n) ways summing to n + 1;
- F(0) + F(1) + ... + F(n − 1) ways summing to n;
- F(0) way summing to −1.
Adding up all n + 1 sums, we see that there are
- (n + 1) F(0) + n F(1) + ... + F(n) ways summing to −1, 0, ..., or n + 1.
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 + 3)
Finally, we complete the proof by subtracting the above identity from n + 1 times the second identity.
 Fifth identity
- The sum of the squares of the first n Fibonacci numbers is the product of the nth and (n + 1)th Fibonacci numbers.
 Identity for doubling n
 Another identity
Another identity useful for calculating Fn for large values of n is
from which other identities for specific values of k, n, and c can be derived below, including
for all integers n and k. Dijkstra points out that doubling identities of this type can be used to calculate Fn using O(log n) long multiplication operations of size n bits. The number of bits of precision needed to perform each multiplication doubles at each step, so the performance is limited by the final multiplication; if the fast Schönhage-Strassen multiplication algorithm is used, this is O(n log n log log n) bit operations. Notice that, with the definition of Fibonacci numbers with negative n given in the introduction, this formula reduces to the double n formula when k = 0.
 Other identities
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.
There are also scaling identities, which take you from Fn and Fn+1 to a variety of things of the form Fan+b; for instance
- 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. Such relations exist in a very general sense for numbers defined by recurrence relations, see the section on multiplication formulae under Perrin numbers for details.
 Power series
This series has a simple and interesting closed-form solution for
This solution can be proven by using the Fibonacci recurrence to expand each coefficient in the infinite sum defining s(x):
Solving the equation s(x) = x + xs(x) + x2s(x) for s(x) results in the closed form solution.
In particular, math puzzle-books note the curious value , or more generally
for all integers k > = 0.
 Reciprocal sums
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 nice nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,
Results such as these make it plausible that a closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but none is yet known. Despite that, the reciprocal Fibonacci constant
Millin series gives a remarkable identity:
which follows from the closed form for its partial sums as N tends to infinity:
 Primes and divisibility
 Fibonacci primes
- 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. They must all have a prime index, except F4 = 3.
144 is the only nontrivial square Fibonacci number. Attila Pethő proved in 2001 that there are only finitely many perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that only 8 and 144 are non-trivial perfect powers.
No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number.
Any three consecutive Fibonacci numbers, taken two at a time, are relatively prime: that is,
- gcd(Fn, Fn+1) = gcd(Fn, Fn+2) = 1.
 Prime divisors of Fibonacci numbers
The divisibility of Fibonacci numbers by a prime p is related to the Legendre symbol which is evaluated as follows:
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:
Examples of all the cases:
For example, F1 = 1, F3 = 2, F5 = 5, F7 = 13, F9 = 34 = 2×17, F11 = 89, F13 = 233, F15 = 610 = 2×5×61
 Divisibility by 11
For example, let n = 1:
F1 + F2 + ... + F10 = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11×13
n = 2:
F2+F3+...+F11 = 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 = 231 = 11×21
n = 3:
F3+F4+...+F12 = 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144= 374 = 11×34
In fact, the identity is true for all integers n, not just positive ones:
n = 0:
F0+F1+...+F9 = 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 = 88 = 11×8
n = −1:
F−1+F0+...+F8 = 1 + 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 = 55 = 11×5
n = −2:
F−2+F−1+F0+...+F7 = −1 + 1 + 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 = 33 = 11×3
 Periodicity modulo n
It is easily seen that if the members of the Fibonacci sequence are taken mod n, the resulting sequence must be periodic with period at most n2. The lengths of the periods for various n form the so-called Pisano periods (sequence A001175 in OEIS). Determining the Pisano periods in general is an open problem, although for any particular n it can be solved as an instance of cycle detection.
 Right triangles
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 Fn, Fn+1, Fn+2 and Fn+3 can also be used to generate a Pythagorean triple in a different way:
Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then:
Example 2: let the Fibonacci numbers be 8, 13, 21 and 34. Then:
 Magnitude of Fibonacci numbers
Since Fn is asymptotic to , the number of digits in is asymptotic to . As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits.
More generally, in the base b representation, the number of digits in Fn is asymptotic to .
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.
Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.
The Fibonacci numbers and principle is also used in the financial markets. It is used in trading algorithms, applications and strategies. Some typical forms include: the Fibonacci fan, the Fibonacci arc, Fibonacci retracement and the Fibonacci time extension.
Fibonacci numbers are used by some pseudorandom number generators.
Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers - by dividing the list so that the two parts have lengths in the approximate proportion φ. A 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 number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods e.g. µ-law.
In music, Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements. It is commonly thought that the first movement of Béla Bartók's Music for Strings, Percussion, and Celesta was structured using Fibonacci numbers.
Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio (denoted φ), the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.
 Fibonacci numbers in nature
Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, 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. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves. The Fibonacci numbers are also found in the family tree of honeybees.
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.
where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j which depends on r, the distance from the center. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.
 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 an egg is laid by an unmated female, it hatches a male.
- If, however, an egg was fertilized by a male, it hatches a female.
Thus, a male bee will always have one parent, and a female bee will have two.
If one traces the ancestry of any male bee (1 bee), he has 1 female parent (1 bee). This female had 2 parents, a male and a female (2 bees). The female had two parents, a male and a female, and the male had one female (3 bees). Those two females each had two parents, and the male had one (5 bees). This sequence of numbers of parents is the Fibonacci sequence.
This is an idealization that does not describe actual bee ancestries. In reality, some ancestors of a particular bee will always be sisters or brothers, thus breaking the lineage of distinct parents.
 Popular culture
The Fibonacci sequence has been generalized in many ways. These 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 in order 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.
- 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.
- Adding other objects than integers, for example functions or strings—one essential example is Fibonacci polynomials.
 See also
|Wikibooks has a book on the topic of|
- Golden ratio
- Lucas number
- Logarithmic spiral
- The Fibonacci Association
- Fibonacci Quarterly — an academic journal devoted to the study of Fibonacci numbers
- Negafibonacci numbers
- Fibonacci word
- Fibonacci heap
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- ^ "Marks for the da Vinci Code: B–". Computer Science For Fun: CS4FN. http://www.cs4fn.org/maths/bee-davinci.php.
- ^ Prusinkiewicz, Przemyslaw; James Hanan (1989). Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics). Springer-Verlag. ISBN 0-387-97092-4.
- ^ Vogel, H (1979), "A better way to construct the sunflower head", Mathematical Biosciences 44 (44): 179–189, doi:10.1016/0025-5564(79)90080-4
- ^ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0387972978. http://algorithmicbotany.org/papers/#webdocs.
- ^ The Fibonacci Numbers and the Ancestry of Bees
- Sequence A000045 Fibonacci Numbers at the On-Line Encyclopedia of Integer Sequences
- Periods of Fibonacci Sequences Mod m at MathPages
- Chaotic behaviour in Fibonacci sequences