A monotonicity property for generalized Fibonacci sequences

2014/10/25 by Mansour, Toufik, Shattuck, Mark
#05A10 #11B39 #11B75 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1410.6943

Abstract

Given k>1, let a_n be the sequence defined by the recurrence a_n=c₁a_n-1+c₂a_n-2+...+c_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 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.

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