Generalizations of Fibonacci numbers

In mathematics, the Fibonacci numbers form a sequence defined recursively by: 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. Using , 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 . See also Negafibonacci coding. 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 for even integers . Similarly, the analytic function: satisfies for odd integers . Finally, putting these together, the analytic function satisfies for all integers . Since for all complex numbers , 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 from the integers to a field for which . These functions are precisely those of the form , so the Fibonacci sequences form a vector space with the functions and as a basis. More generally, the range of 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. The 2-dimensional -module of Fibonacci integer sequences consists of all integer sequences satisfying . Expressed in terms of two initial values we have: where 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 .