In mathematics, the Fibonacci numbers form a sequence defined recursively by:
 F_{0} = 0
 F_{1} = 1
 F_{n} = F_{n − 1} + F_{n − 2}, for integer n > 1.
That is, after two starting values, each number is the sum of the two preceding numbers.
The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers.
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Transcription
Contents
Extension to negative integers
Using F_{n − 2} = F_{n} − F_{n − 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)^{n + 1}F_{n}.
See also Negafibonacci ^{[1]} .
Extension to all real or complex numbers
There are a number of possible generalizations of the Fibonacci numbers which include the real numbers (and sometimes the complex numbers) in their domain. These each involve the golden ratio φ, and are based on Binet's formula
The analytic function
has the property that Fe(n) = F_{n} for even integers n.^{[2]} Similarly, the analytic function:
satisfies Fo(n) = F_{n} for odd integers n.
Finally, putting these together, the analytic function
satisfies Fib(n) = F_{n} for all integers n.^{[3]}
Since Fib(z + 2) = Fib(z + 1) + Fib(z) for all complex numbers z, this function also provides an extension of the Fibonacci sequence to the entire complex plane. Hence we can calculate the generalized Fibonacci function of a complex variable, for example,
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 Zmodule). Then the Fibonacci sequences form a 2dimensional Zmodule in the same way.
Similar integer sequences
Fibonacci integer sequences
The 2dimensional Zmodule of Fibonacci integer sequences consists of all integer sequences satisfying g(n + 2) = g(n) + g(n + 1). Expressed in terms of two initial values we have:
where φ is the golden ratio.
The ratio between two consecutive elements converges to the golden ratio, except in the case of the sequence which is constantly zero and the sequences where the ratio of the two first terms is (−φ)^{−1}.
The sequence can be written in the form
in which a = 0 if and only if b = 0. In this form the simplest nontrivial example has a = b = 1, which is the sequence of Lucas numbers:
We have L_{1} = 1 and L_{2} = 3. The properties include:
Every nontrivial Fibonacci integer sequence appears (possibly after a shift by a finite number of positions) as one of the rows of the Wythoff array. The Fibonacci sequence itself is the first row, and a shift of the Lucas sequence is the second row.^{[4]}
See also Fibonacci integer sequences modulo n.
Lucas sequences
A different generalization of the Fibonacci sequence is the Lucas sequences of the kind defined as follows:
 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.
When Q = −1, this sequence is called PFibonacci sequence, for example, Pell sequence is also called 2Fibonacci sequence.
The 3Fibonacci sequence is
 0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, ... (sequence A006190 in the OEIS)
The 4Fibonacci sequence is
 0, 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289, 7465176, 31622993, 133957148, 567451585, 2403763488, ... (sequence A001076 in the OEIS)
The 5Fibonacci sequence is
 0, 1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285, 9989688826, ... (sequence A052918 in the OEIS)
The 6Fibonacci sequence is
 0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202, ... (sequence A005668 in the OEIS)
The nFibonacci constant is the ratio toward which adjacent nFibonacci numbers tend; it is also called the nth metallic mean, and it is the only positive root of x^{2} − nx − 1 = 0. For example, the case of n = 1 is 1 + √5/2, or the golden ratio, and the case of n = 2 is 1 + √2, or the silver ratio. Generally, the case of n is n + √n^{2}+4/2.^{[citation needed]}
Generally, U(n) can be called (P,−Q)Fibonacci sequence, and V(n) can be called (P,−Q)Lucas sequence.
The (1,2)Fibonacci sequence is
 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, ... (sequence A001045 in the OEIS)
The (1,3)Fibonacci sequence is
 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, 14209, 32689, 75316, 173383, 399331, 919480, 2117473, 4875913, 11228332, 25856071, 59541067, ... (sequence A006130 in the OEIS)
The (2,2)Fibonacci sequence is
 0, 1, 2, 6, 16, 44, 120, 328, 896, 2448, 6688, 18272, 49920, 136384, 372608, 1017984, 2781184, 7598336, 20759040, 56714752, ... (sequence A002605 in the OEIS)
The (3,3)Fibonacci sequence is
 0, 1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893, 507627, 1924560, 7296561, 27663363, 104879772, 397629405, 1507527531, 5715470808, ... (sequence A030195 in the OEIS)
Fibonacci numbers of higher order
A Fibonacci sequence of order n is an integer sequence in which each sequence element is the sum of the previous n elements (with the exception of the first n elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2. The cases n = 3 and n = 4 have been thoroughly investigated. The number of compositions of nonnegative integers into parts that are at most n is a Fibonacci sequence of order n. The sequence of the number of strings of 0s and 1s of length m that contain at most n consecutive 0s is also a Fibonacci sequence of order n.
These sequences, their limiting ratios, and the limit of these limiting ratios, were investigated by Mark Barr in 1913.^{[5]}
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, … (sequence A000073 in the OEIS)
The tribonacci constant
is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x^{3} − x^{2} − x − 1 = 0, and also satisfies the equation x + x^{−3} = 2. It is important in the study of the snub cube.
The reciprocal of the tribonacci constant, expressed by this relation ξ^{3} + ξ^{2} + ξ = 1, can be written as:
The tribonacci numbers are also given by^{[6]}
where ⌊ · ⌉ denotes the nearest integer function and
 .
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, … (sequence A000078 in the OEIS)
The tetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is a root of the polynomial x^{4} − x^{3} − x^{2} − x − 1 = 0, approximately 1.927561975482925 OEIS: A086088, and also satisfies the equation x + x^{−4} = 2.
The tetranacci constant is expressed in terms of radicals by^{[7]}
where
Higher orders
Pentanacci, hexanacci, and heptanacci numbers have been computed. The pentanacci numbers are:
 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, … (sequence A001591 in the OEIS)
Hexanacci numbers:
 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, … (sequence A001592 in the OEIS)
Heptanacci numbers:
 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, … (sequence A122189 in the OEIS)
Octanacci numbers:
 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, ... (sequence A079262 in the OEIS)
Enneanacci numbers:
 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, ... (sequence A104144 in the OEIS)
The limit of the ratio of successive terms of an nnacci series tends to a root of the equation x + x^{−n} = 2 (OEIS: A103814, OEIS: A118427, OEIS: A118428).
An alternate recursive formula for the limit of ratio r of two consecutive nnacci numbers can be expressed as
 .
The special case n = 2 is the traditional Fibonacci series yielding the golden section φ = 1 + 1/φ.
The above formulas for the ratio hold even for nnacci series generated from arbitrary numbers. The limit of this ratio is 2 as n increases. An "infinacci" 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 the powers of two.
The limit of the ratio for any n > 0 is the positive root r of the characteristic equation^{[7]}
The root r is in the interval 2(1 − 2^{−n}) < r < 2. The negative root of the characteristic equation is in the interval (−1, 0) when n is even. This root and each complex root of the characteristic equation has modulus 3^{−n} < r < 1.^{[7]}
A series for the positive root r for any n > 0 is^{[7]}
There is no solution of the characteristic equation in terms of radicals when 5 ≤ n ≤ 11. ^{[7]}
The kth element of the nnacci sequence is given by
where ⌊ · ⌉ denotes the nearest integer function and r is the nnacci constant, which is the root of x + x^{−n} = 2 nearest to 2.^{[8]}
A cointossing problem is related to the nnacci sequence. The probability that no n consecutive tails will occur in m tosses of an idealized coin is 1/2^{m}F^{(n)}
_{m + 2}.^{[9]}
Fibonacci word
In analogy to its numerical counterpart, the Fibonacci word is defined by:
where + denotes the concatenation of two strings. The sequence of Fibonacci strings starts:
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.
If "a" and "b" represent two different materials or atomic bond lengths, the structure corresponding to a Fibonacci string is a Fibonacci quasicrystal, an aperiodic quasicrystal structure with unusual spectral properties.
Convolved Fibonacci sequences
A convolved Fibonacci sequence is obtained applying a convolution operation to the Fibonacci sequence one or more times. Specifically, define^{[10]}
and
The first few sequences are
 r = 1: 0, 0, 1, 2, 5, 10, 20, 38, 71, … (sequence A001629 in the OEIS).
 r = 2: 0, 0, 0, 1, 3, 9, 22, 51, 111, … (sequence A001628 in the OEIS).
 r = 3: 0, 0, 0, 0, 1, 4, 14, 40, 105, … (sequence A001872 in the OEIS).
The sequences can be calculated using the recurrence
The generating function of the rth convolution is
The sequences are related to the sequence of Fibonacci polynomials by the relation
where F^{(r)}
_{n}(x) is the rth derivative of F_{n}(x). Equivalently, F^{(r)}
_{n} is the coefficient of (x − 1)^{r} when F_{n}(x) is expanded in powers of (x − 1).
The first convolution, F^{(1)}
_{n} can be written in terms of the Fibonacci and Lucas numbers as
and follows the recurrence
Similar expressions can be found for r > 1 with increasing complexity as r increases. The numbers F^{(1)}
_{n} are the row sums of Hosoya's triangle.
As with Fibonacci numbers, there are several combinatorial interpretations of these sequences. For example F^{(1)}
_{n} is the number of ways n − 2 can be written as an ordered sum involving only 0, 1, and 2 with 0 used exactly once. In particular F^{(1)}
_{4} = 5 and 2 can be written 0 + 1 + 1, 0 + 2, 1 + 0 + 1, 1 + 1 + 0, 2 + 0.^{[11]}
Other generalizations
The Fibonacci polynomials are another generalization of Fibonacci numbers.
The Padovan sequence is generated by the recurrence P(n) = P(n − 2) + P(n − 3).
The Narayana's cows sequence is generated by the recurrence N(k) = N(k − 1) + N(k − 3).
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 Furstenberg 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 such 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, … (sequence A007629 in the OEIS)
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 twodimensional. 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 2, 1, 3, 4, 7, 11, ..., then we obtain
 L(n) = 2F(n − 1) + F(n).
Ngenerated Fibonacci sequence
We can define the Ngenerated Fibonacci sequence (where N is a positive rational number): if
where p_{r} is the rth prime, then we define
If n = r − 1, then F_{N}(n) = 1, and if n < r − 1, then F_{N}(n) = 0.^{[citation needed]}
Sequence N OEIS sequence Fibonacci sequence 6 A000045 Pell sequence 12 A000129 Jacobsthal sequence 18 A001045 Tribonacci sequence 30 A000073 Tetranacci sequence 210 A000288 Padovan sequence 15 A000931 Narayana's cows sequence 10 A000930
SemiFibonacci sequence
The semiFibonacci sequence (sequence A030067 in the OEIS) is defined via the same recursion for oddindexed terms and , but for even indices , . The bissection A030068 of oddindexed terms therefore verifies and is strictly increasing. It yields the set of the semiFibonacci numbers
 1, 2, 3, 5, 6, 9, 11, 16, 17, 23, 26, 35, 37, 48, 53, 69, 70, 87, 93, 116, 119, 145, 154, ... (sequence A030068 in the OEIS)
which occur as .
References
 ^ Triana, Juan. Negafibonacci numbers via matrices. Bulletin of TICMI, 2019, págs. 1924.
 ^ What is a Fibonacci Number ?
 ^ Pravin Chandra and Eric W. Weisstein. "Fibonacci Number". MathWorld.
 ^ Morrison, D. R. (1980), "A Stolarsky array of Wythoff pairs", A Collection of Manuscripts Related to the Fibonacci Sequence (PDF), Santa Clara, CA: The Fibonacci Association, pp. 134–136.
 ^ Gardner, Martin (1961). The Scientific American Book of Mathematical Puzzles and Diversions, Vol. II. Simon and Schuster. p. 101.
 ^ Simon Plouffe, 1993
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Wolfram, D.A. (1998). "Solving Generalized Fibonacci Recurrences" (PDF). Fib. Quart.
 ^ Du, Zhao Hui, 2008
 ^ Eric W. Weisstein. "Coin Tossing". MathWorld.
 ^ V. E. Hoggatt, Jr. and M. BicknellJohnson, "Fibonacci Convolution Sequences", Fib. Quart., 15 (1977), pp. 117122.
 ^ Sloane, N. J. A. (ed.). "Sequence A001629". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation.
External links
 Hazewinkel, Michiel, ed. (2001) [1994], "Tribonacci number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104