Generalised Fibonacci sequences constructed from balancing words

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.