Define an integer sequence (A_n)_n≥0 by setting A₀ = a, A₁ = b, and A_n+1 = pA_n + qA_n–1 for all n. We consider the case q = 1 to explore the problem of finding all rational numbers x such that the generating function of (A_n) yields an integer when evaluated at x. We point out that we can divide the set of all x-values into families and find some families that always exist. Then we provide an algorithm to find all the families through a finite computation. Finally, we apply the algorithm to the special cases that (a,b) = (0,1) and (a,b) = (1,1).