A monotonicity property for generalized Fibonacci sequences

Given k ≥ 2, let a_n be the sequence defined by the recurrence a_n = α₁a_n-1 + ⋯ + α_ka_n-k for n ≥ k, with initial values a₀ = a₁ = ⋯ = a_k-2 = 0 and a_k-1 = 1. We show under a couple of assumptions concerning the constants α_i that the ratio annan-1n-1 n√an/n-1√an-1 is strictly decreasing for all n ≥ N, for some N depending on the sequence, and has limit 1. In particular, this holds in the cases when all of the α_i are unity or when all of the α_i are zero except for the first and last, which are unity. Furthermore, when k = 3 or k = 4, it is shown that one may take N to be an integer less than 12 in each of these cases.