On the limit of a recurrence relation

In this paper we study the asymptotic properties of the sequence of integers g(n), defined by the following recurrence relation: g(n + 1) = {[}(1 + alpha/n-alpha)g(n)], where alpha > 0 and {[}x] denotes the largest integer not greater than x. For any alpha > 0, the limit g(n)/n(alpha) exists. We prove that for alpha = 2, this limit is always rational. For alpha = 3, we give some sufficient conditions which guarantee that the limit is rational.