Generalizations of Fibonacci numbers

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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 other objects than numbers.

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[edit] 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.

[edit] 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.

[edit] Similar integer sequences

[edit] 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

\left( \frac 1 2 \left( 1 + \sqrt{5} \right) \right)^n = \frac 1 2 \left( L(n) + F(n) \sqrt{5} \right).

Lucas numbers are related to Fibonacci numbers by the relation

L\left(n\right)=F\left(n-1\right)+F\left(n+1\right).\,

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).

[edit] 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 A000073:

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

T(n) = \left[ 3 \, b \frac{\left(\frac{1}{3} \left( a_{+} + a_{-} + 1\right)\right)^n}{b^2-2b+4} \right]

where the outer brackets denote the nearest integer function and

a_{\pm} = \left(19 \pm 3 \sqrt{33}\right)^{1/3}
b = \left(586 + 102 \sqrt{33}\right)^{1/3}

(Simon Plouffe, 1993).[1]

[edit] 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 A000078:

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.

[edit] Other -nacci numbers

Pentanacci, hexanacci and heptanacci numbers have been computed, with perhaps less interest so far in research.[citation needed]

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.

[edit] Fibonacci strings

In analogy to its numerical counterpart, a Fibonacci string is defined by:

F_n := F(n):= \begin{cases} b & \mbox{if } n = 0; \\ a & \mbox{if } n = 1; \\ F(n-1)+F(n-2) & \mbox{if } n > 1. \\ \end{cases},

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.

[edit] 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 A007629:

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 2, 1, 3, 4, 7, 11…, then we obtain L(n) = 2F(n − 1) + F(n).

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