A monotonicity property for generalized Fibonacci sequences

Abstract

Given k>1, let a_n be the sequence defined by the recurrence a_n=c_1a_{n-1}+c_2a_{n-2}+...+c_ka_{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 c_i that the ratio of the n-th root of a_n to the (n-1)-st root of 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 c_i are unity or when all of the c_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.