Given k ≥ 2, let an be the sequence defined by the recurrence an = α1an–1 + … + αkan–k for n ≥ k, with initial values a0 = a1 = … = ak–2 = 0 and ak–1 = 1. We show under a couple of assumptions concerning the constants αi that the ratio 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 αi are unity or when all of the αi 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.
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