Generalizations of Fibonacci numbers
, one can extend the Fibonacci numbers to negative integers.
These each involve the golden ratio φ, and are based on Binet's formula The analytic function has the property that
Finally, putting these together, the analytic function satisfies
, 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, The term Fibonacci sequence is also applied more generally to any function
, so the Fibonacci sequences form a vector space with the functions
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.
Such sequences have applications in number theory and primality proving.
-Fibonacci numbers tend; it is also called the nth metallic mean, and it is the only positive root of
The number of compositions of nonnegative integers into parts that are at most
The first few tribonacci numbers are: The series was first described formally by Agronomof[6] in 1914,[7] but its first unintentional use is in the Origin of Species by Charles R. Darwin.
In the example of illustrating the growth of elephant population, he relied on the calculations made by his son, George H.
The reciprocal of the tribonacci constant, expressed by the relation
, approximately 1.927561975482925 (sequence A086088 in the OEIS), and also satisfies the equation
Pentanacci, hexanacci, heptanacci, octanacci and enneanacci numbers have been computed.
, approximately 1.965948236645485 (sequence A103814 in the OEIS), and also satisfies the equation
An alternate recursive formula for the limit of ratio
is the traditional Fibonacci series yielding the golden section
The negative root of the characteristic equation is in the interval (−1, 0) when
is[11] There is no solution of the characteristic equation in terms of radicals when 5 ≤ n ≤ 11.
[12] In analogy to its numerical counterpart, the Fibonacci word is defined by: where
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.
can be written in terms of the Fibonacci and Lucas numbers as and follows the recurrence Similar expressions can be found for
The Narayana's cows sequence is generated by the recurrence
A random Fibonacci sequence can be defined by tossing a coin for each position
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.
Any such sequence is uniquely determined by a choice of two elements, so the vector space is two-dimensional.
are seen to form a canonical basis for this space, yielding the identity: for all such sequences S. For example, if S is the Lucas sequence 2, 1, 3, 4, 7, 11, ..., then we obtain We can define the N-generated Fibonacci sequence (where N is a positive rational number): if where pr is the rth prime, then we define If
It yields the set of the semi-Fibonacci numbers which occur as
