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The Fibonacci numbers are the sequence of numbers ${F_n}_(n=1)^infty$ defined by the linear recurrence equation

	(1)

with F_1=F_2=1 . As a result of the definition (1), it is conventional to define F_0=0 .

The Fibonacci numbers for n=1 , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... (OEIS A000045).

Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials F_n(x) with F_n=F_n(1) .

Fibonacci numbers are implemented in the Wolfram Language as Fibonacci[_n_].

The Fibonacci numbers are also a Lucas sequence U_n(1,-1) , and are companions to the Lucas numbers (which satisfy the same recurrence equation).

FoxTrot by Bill Amend

The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). (The right panel instead applies the Perrin sequence).

A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 43, 60-61, and 189-192). In the Season 1 episode "Sabotage" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that the Fibonacci numbers are found in the structure of crystals and the spiral of galaxies and a nautilus shell. In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.

Binary plot of the Fibonacci sequence

The plot above shows the first 511 terms of the Fibonacci sequence represented in binary, revealing an interesting pattern of hollow and filled triangles (Pegg 2003). A fractal-like series of white triangles appears on the bottom edge, due in part to the fact that the binary representation of F_(2^n+2^(n+1)) ends in n+2 zeros. Many other similar properties exist.

The Fibonacci numbers give the number of pairs of rabbits months after a single pair begins breeding (and newly born bunnies are assumed to begin breeding when they are two months old), as first described by Leonardo of Pisa (also known as Fibonacci) in his book Liber Abaci. Kepler also described the Fibonacci numbers (Kepler 1966; Wells 1986, pp. 61-62 and 65). Before Fibonacci wrote his work, the Fibonacci numbers had already been discussed by Indian scholars such as Gopāla (before 1135) and Hemachandra (c. 1150) who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes or syllables. The number of such rhythms having beats altogether is F_(n+1) , and hence these scholars both mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly (Knuth 1997, p. 80).

The numbers of Fibonacci numbers less than 10, 10^2 , 10^3 , ... are 6, 11, 16, 20, 25, 30, 35, 39, 44, ... (OEIS A072353). For n=1 , 2, ..., the numbers of decimal digits in F_(10^n) are 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, ... (OEIS A068070). As can be seen, the initial strings of digits settle down to produce the number 208987640249978733769..., which corresponds to the decimal digits of lnphi=0.2089876... (OEIS A097348), where phi is the golden ratio. This follows from the fact that for any power function f_n=c^n , the number of decimal digits for f_(10^n) is given by 10^nlog_(10)c .

The Fibonacci numbers F_n , are squareful for n=6 , 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ..., 372, 375, 378, 384, ... (OEIS A037917) andsquarefree for n=1 , 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (OEIS A037918). 4|F_(6n) and 25|F_(25n) for all , and there is at least one n<=2m such that m|F_n . No squareful Fibonacci numbers F_p are known with prime.

The ratios of successive Fibonacci numbers F_n/F_(n-1) approaches the golden ratio phi as approaches infinity, as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62). The ratios of alternate Fibonacci numbers are given by the convergents to phi^(-2) , where phi is the golden ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant (phyllotaxis): for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). The Fibonacci numbers are sometimes called pine cone numbers (Pappas 1989, p. 224). The role of the Fibonacci numbers in botany is sometimes called Ludwig's law (Szymkiewicz 1928; Wells 1986, p. 66; Steinhaus 1999, p. 299). However, botanist Cooke suggests caution in making correlations between botany and the Fibonacci sequence (Peterson 2006).

The equation (◇) is a linear recurrence equation

	(2)

so the closed form for F_n is given by

	(3)

where alpha and beta are the roots of x^2=Ax+B . Here, A=B=1 , so the equation becomes

	(4)

which has roots

	(5)

The closed form is therefore given by

	(6)

This is known as Binet's formula (Wells 1986, p. 62). Another closed form is

where [x] is the nearest integer function (Wells 1986, p. 62).

Fibonacci

Using equation (7), the definition of F_n can be extended to negative integers according to

	(9)

More generally, the Fibonacci numbers can be extended to a real number via

$F_nu=1/(sqrt(5)){((1+sqrt(5))/2)^nu-(2/(1+sqrt(5)))^nucos(nupi)},$	(10)

as plotted above.

FibonacciRoots

The Fibonacci function has zeros at x=0 and an infinite number of negative values that approach n+0.5 for all negative integers , given by the solutions to

	(11)

where phi is the golden ratio. The first few roots are 0, x=-0.183802... (OEIS A089260), -1.5707764... , -2.4704268... , -3.5108513... , ....

Another recurrence relation for the Fibonacci numbers is

| $F_(n+1)=\|_(F_n(1+sqrt(5))+1)/2_|=|_phiF_n+1/2_|,$ | (12) | | --------------------------------------------------------------------------------------------------------------------------------------------- | ---- |

where |_x_| is the floor function and phi is the golden ratio. This expression follows from the more general recurrence relation

| $\|F_(n+1) F_(n+2) ... F_(n+k); F_(n+k+1) F_(n+k+2) ... F_(n+2k); | | ... | ; F_(n+k(k-1)+1) F_(n+k(k-1)+2) ... F_(n+k^2)|=0$ | (13) | | ------------------------------------------------------------------- | | --- | ----------------------------------------------------------------------------------------------------------------------------------------- | ---- |

for k>2 . (The k=1 case is trivially F_(n+1) , while the k=2 case is essentially Cassini's identity and therefore equal to (-1)^n .)

Another interesting determinant identity follows from defining A_n as the n×n matrix with zeros everywhere except a_(ii)=1 and a_(ij)=sqrt(-1) for |i-j|=1 (i.e., along the superdiagonal and subdiagonal). Then

| $\|A_n|=F_(n+1)$ | (14) | | ---------------------------------------------------------------------------------------------------------- | ---- |

(S. Markelov).

The generating function for the Fibonacci numbers is

FibonacciSlash

By plugging in x=1/10 , this gives the curious addition tree illustrated above,

	(18)

	(19)

(Livio 2002, pp. 106-107).

The sum

	(20)

(OEIS A079586) is known as the reciprocal Fibonacci constant.

Yuri Matiyasevich (1970) showed that there is a polynomial in , , and a number of other variables , , , ... having the property that n=F_(2m) iff there exist integers , , , ... such that P(n,m,x,y,z,...)=0 . This led to the proof of the impossibility of the tenth of Hilbert's problems (does there exist a general method for solving Diophantine equations?) by Julia Robinson and Martin Davis in 1970 (Reid 1997, p. 107).

FibonacciChecker3

FibonacciChecker4

FibonacciChecker5

The Fibonacci number F_(n+1) gives the number of ways for 2×1 dominoes to cover a 2×n checkerboard, as illustrated in the diagrams above (Dickau).

The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers is F_(n+2) . The number of ways of picking a set (including theempty set) from the numbers 1, 2, ..., without picking two consecutive numbers (where 1 and are now consecutive) is L_n=F_(n+1)+F_(n-1) , where L_n is a Lucas number.

The probability of not getting two heads in a row in tosses of a coin is F_(n+2)/2^n (Honsberger 1985, pp. 120-122). Fibonacci numbers are also related to the number of ways in which coin tosses can be made such that there are not three consecutive heads or tails. The number of ideals of an -element fence poset is the Fibonacci number F_n .

Given a resistor network of 1- Omega resistors, each incrementally connected in series or parallel to the preceding resistors, then the net resistance is a rational number having maximum possible denominator of F_(n+1) .

The Fibonacci numbers are given in terms of the Chebyshev polynomial of the second kind by

	(21)

Sum identities include

There are a number of particular pretty algebraic identities involving the Fibonacci numbers, including

(Brousseau 1972), Catalan's identity

	(32)

d'Ocagne's identity

	(33)

and the Gelin-Cesàro identity

	(34)

Letting r=1 in (32) gives Cassini's identity

	(35)

sometimes also called Simson's formula since it was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165-168; Petkovšek et al. 1996, p. 12).

Johnson (2003) gives the very general identity

	(36)

which holds for arbitrary integers , , , , and with a+b=c+d and from which many other identities follow as special cases.

The Fibonacci numbers obey the negation formula

	(37)

the addition formula

	(38)

where L_n is a Lucas number, the subtraction formula

	(39)

the fundamental identity

	(40)

conjugation relation

	(41)

successor relation

	(42)

double-angle formula

	(43)

multiple-angle recurrence

	(44)

multiple-angle formulas

(where (48) holds only for n>1 ), the extension

	(50)

(A. Mihailovs, pers. comm., Jan. 24, 2003), product expansions

	(51)

and

	(52)

square expansion,

	(53)

and power expansion

| $F_n^k=1/(2·5^(\|_k/2_|))sum_(i=0)^k(k; i)(-1)^(i(n+1)){F_((k-2i)n) for k odd; L_((k-2i)n) for k even.$ | (54) | | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---- |

Honsberger (1985, p. 107) gives the general relations

In the case l=n-l+1 , then l=(n+1)/2 and for odd,

	(58)

Similarly, for even,

	(59)

Letting k=(n-1)/2 gives the identities

FibonacciShallowDiags

Sum formulas for F_n include

(Wells 1986, p. 63), the latter of which shows that the shallow diagonals" of Pascal's triangle sum to Fibonacci numbers (Pappas 1989). Additional identities can be found throughout the Fibonacci Quarterly journal. A list of 47 generalized identities are given by Halton (1965).

In terms of the Lucas number L_n ,

(Honsberger 1985, pp. 111-113). A remarkable identity is

	(69)

(Honsberger 1985, pp. 118-119), which can be generalized to

	(70)

(Johnson 2003). It is also true that

	(71)

for odd, and

	(72)

for even (Freitag 1996).

From (◇), the ratio of consecutive terms is

which is just the first few terms of the continued fraction for the golden ratio phi . Therefore,

	(78)

Another fascinating connection with the golden ratiois given by the series

	(79)

Guy (1990) notes the curious fact that [e^((n-1)/2)] for n=0 , 1, ... gives 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ..., but then continues 91, 149, ... (OEIS A005181).

Taking the product of the first Fibonacci numbers and adding 1 for n=1 , 2, ... gives the sequence 2, 2, 3, 7, 31, 241, ... (OEISA052449). Of these, 2, 2, 3, 7, 31, 241, 3121, ... (OEIS A053413) are prime, i.e., the terms 1, 2, 3, 4, 5, 6, 7, 8, 22, 28, ... (OEIS A053408).

The sequence of final digits in Fibonacci numbers repeats in cycles of 60. The last two digits repeat in 300, the last three in 1500, the last four in 15000 , etc. The number of Fibonacci numbers between and is either 1 or 2 (Wells 1986, p. 65).

Cesàro derived the finite sums

(Honsberger 1985, pp. 109-110). The Fibonacci numbers satisfy the power recurrence

	(82)

where [a; b]_F is a Fibonomial coefficient, the reciprocal sum

	(83)

the convolution

	(84)

the partial fraction decomposition

	(85)

where

and the summation formula

	(89)

where

	(90)

Infinite sums include

	(91)

(Clark 1995) and

where phi is the golden ratio (Wells 1986, p. 65).

For n>=3 , F_n|F_m iff n|m (Wells 1986, p. 65). L_n|L_m iff divides into an odd number of times. (F_m,F_n)=F_((m,n)) (Michael 1964; Honsberger 1985, pp. 131-132). No odd Fibonacci number is divisible by 17 (Honsberger 1985, pp. 132 and 242). No Fibonacci number is ever of the form p-1 or p+1 where is a prime number (Honsberger 1985, p. 133).

Consider the sum

This is a telescoping sum, so

	(96)

thus

	(97)

(Honsberger 1985, pp. 134-135). Using Binet's formula, it also follows that

	(98)

where

	(101)

	(102)

(Honsberger 1985, pp. 138 and 242-243). The Millin series has sum

	(103)

(Honsberger 1985, pp. 135-137).

The Fibonacci numbers are complete. In fact, dropping one number still leaves a complete sequence, although dropping two numbers does not (Honsberger 1985, pp. 123 and 126). Dropping two terms from the Fibonacci numbers produces a sequence which is not even weakly complete (Honsberger 1985, p. 128). However, the sequence

	(104)

is weakly complete, even with any finite subsequence deleted (Graham 1964). ${F_n^2}$ is not complete, but ${F_n^2}+{F_n^2}$ are. 2^(N-1) copies of ${F_n^N}$ are complete.

For a discussion of square Fibonacci numbers, see Cohn (1964ab), who proved that the only square number Fibonacci numbers are 1 and F_(12)=144 (Cohn 1964ab, Guy 1994). Ming (1989) proved that the only triangular Fibonacci numbers are 1, 3, 21, and 55. The Fibonacci and Lucas numbers have no common terms except 1 and 3. The only cubic Fibonacci numbers are 1 and 8.

	(105)

is a Pythagorean triple, as first discovered by Raine (Livio 2002, p. 107).

	(106)

is always a square number (Honsberger 1985, p. 243).

In 1975, James P. Jones showed that the Fibonacci numbers are the positive integer values of the polynomial

	(107)

for Gaussian integers and (Le Lionnais 1983). If and are two positive integers, then between n^k and n^(k+1) , there can never occur more than Fibonacci numbers (Honsberger 1985, pp. 104-105).

The Fibonacci numbers satisfy the identity

	(108)

where GCD(a,b) is the greatest common divisor.

The sequence of Fibonacci numbers is periodic modulo any modulus (Wall 1960). These periods are known as Pisano periods pi(m) (Wrench 1969). The Fibonacci numbers modulo for small are tabulated below, together with their Pisano periods.

		OEIS	${F_n}$ (mod )
2	3	A011655	1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...
3	8	A082115	1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, ...
4	6	A079343	1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, ...
5	20	A082116	1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, ...
6	24	A082117	1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, ...
7	16	A082116	1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, ...
8	12	A079344	1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1, 0, 1, 1, 2, ...

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References

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