A monotonicity property for generalized Fibonacci sequences

TL;DR

This paper proves a monotonicity property of the ratio of consecutive roots in generalized Fibonacci sequences, establishing conditions under which this ratio decreases and converges to 1, with specific bounds for small sequence orders.

Contribution

It introduces new conditions ensuring the decreasing nature of the root ratios in generalized Fibonacci sequences and provides explicit bounds for small sequence orders.

Findings

The ratio of the n-th root to the (n-1)-th root is strictly decreasing under certain conditions.

The ratio converges to 1 as n approaches infinity.

For k=3 or 4, the decreasing property holds from n greater than 12.

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.