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) = Fn for even integers n. Similarly, the analytic function:
satisfies Fo(n) = Fn for odd integers n.
Finally, putting these together, the analytic function
satisfies Fib(n)=Fn for all integers n.
This formula can be used to compute the generalized Fibonacci function of a complex variable. For example,
Read more about this topic: Generalizations Of Fibonacci Numbers
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