A monotonicity property for generalized Fibonacci sequences

Abstract

Abstract Given k ≥ 2, let a n be the sequence defined by the recurrence a n = α 1 a n–1 + … + α k a n–k for n ≥ k, with initial values a 0 = a 1 = … = a k–2 = 0 and a k–1 = 1. We show under a couple of assumptions concerning the constants α i that the ratio a n n a n − 1 n − 1 $\frac{\sqrt[n]{a_n}}{\sqrt[n-1]{a_{n-1}}}$ 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.