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.

1 Extension to negative integers
2 Extension to all real or complex numbers
3 Vector space
4 Similar integer sequences
5 Fibonacci strings
6 Other generalizations
7 See also
8 References

[edit] 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)ⁿF_n.

[edit] 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

$F_n = \frac{\phi^n-(1-\phi)^n}{\sqrt{5}}.$

The analytic function

$Fe(x) = \frac{\phi^x - \phi^{-x}}{\sqrt{5}}$

has the property that Fe(n) = F_n for even integers n. ^[1] Similarly, the analytic function:

$Fo(x) = \frac{\phi^x + \phi^{-x}}{\sqrt{5}}$

satisfies Fo(n) = F_n for odd integers n.

Finally, putting these together, the analytic function

$Fib(x) = \frac{\phi^x - \cos(x \pi)\phi^{-x}}{\sqrt{5}}$

satisfies Fib(n)=F_n for all integers n. ^[2]

This formula can be used to compute the generalized Fibonacci function of a complex variable. For example,

$Fib(3+4i) \approx -5248.5 - 14195.9 i$

[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] Fibonacci integer sequences

The 2-dimensional Z-module 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:

g(n) = F(n)g(1) + F(n-1)g(0) = $g(1){{\varphi^n-(-\varphi)^{-n}} \over {\sqrt 5}}+g(0){{\varphi^{n-1}-(-\varphi)^{1-n}} \over {\sqrt 5}}\, ,$

where $\varphi$ is the golden ratio.

Written in the form

$a\varphi^n+b(-\varphi)^{-n}$

a = 0 if and only if b = 0.

Thus the ratio between two consecutive elements converges to the golden ratio, except in the case of the sequence which is constantly zero.

Written in this form the simplest non-trivial example (a = b = 1) is the sequence of Lucas numbers:

$L_n = \varphi^n + (- \varphi)^{- n}$

We have L(1) = 1 and L(2) = 3. The properties include:

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

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

[edit] Lucas sequences

A generalization of the Fibonacci sequence are 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.

[edit] 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.

[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 $\frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3}$ 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

$a_{\pm} = \left(19 \pm 3 \sqrt{33}\right)^{1/3}$

$b = \left(586 + 102 \sqrt{33}\right)^{1/3}$ . ^[3]

[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] Higher orders

Pentanacci, hexanacci and heptanacci numbers have been computed, with perhaps less interest so far in research.^{[citation needed]} The pentanacci numbers are:

0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, ... (A001591).

Hexanacci numbers:

0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, ... (A001592).

Heptanacci numbers:

0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, ... (A122189).

In the limit, the ratio of successive terms of an n-nacci sequence tends to a root of the equation $x+x^{-n}=2\,$ .

Thus, this has a limit as n increases. 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.

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.

[edit] 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).

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 $L (n) = 2 F (n - 1) + F (n)$ .

[edit] See also

Fibonacci family

[edit] References

^ What is a Fibonacci Number ?
^ Pravin Chandra and Eric W. Weisstein, Fibonacci Number at MathWorld.
^ Simon Plouffe, 1993

Categories: Fibonacci numbers

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