Generalised Fibonacci sequences constructed from balancing words

TL;DR

This paper extends previous results on the growth rates of generalized Fibonacci sequences constructed with patterned plus-minus rules, analyzing sequences defined by balancing words and variable coefficients.

Contribution

It generalizes McLellan's 2012 results to sequences with variable coefficients and patterned plus-minus rules based on balancing words.

Findings

Established existence of growth rate limits for these sequences.

Extended analysis to variable coefficient recurrences.

Provided conditions under which the limits exist.

Abstract

We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference of the two preceding terms where the pluses and minuses follow a certain pattern. In 2012, McLellan proved that if the pluses and minuses follow a periodic pattern and $G_n$ is the $n$th term of the resulting generalised Fibonacci sequence, then \begin{equation*} \lim_{n\rightarrow\infty}|G_n|^{1/n} \end{equation*} exists. We extend her results to recurrences of the form $G_{m+2} = \alpha_m G_{m+1} \pm G_{m}$ if the choices of pluses and minuses, and of the $\alpha_m$ follow a balancing word type pattern.