Fibonacci Generating Functions

Define an integer sequence (G(n))(n is an element of Z) by setting G(0) = a, G(1) = b, and G(n) = G(n-1) + G(n-2) for all n. In this paper, we explore the problem of finding all rational numbers x such that the generating function of the sequence yields an integer when evaluated at x. We show that these numbers can be naturally divided into families and find some families that are always present. Then we give an algorithm that, for each choice of a and b, reduces the problem of finding all of the families to a finite computation.