Generalised Fibonacci sequences constructed from balanced 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 nth term of the resulting generalised Fibonacci sequence, then limn→∞⁡|G_n|^1/n exists. We extend her results to recurrences of the form G_m+2=α_mG_m+1±G_m if the choices of pluses and minuses, and of the α_m follow a balancing word type pattern.