gr.group theory - On the size of balls in Cayley graphs

In the article

R. Grigorchuk and P. De La Harpe, On problems related to growth, entropy, and spectrum in group theory, Journal of Dynamical and Control Systems, Volume 3, Number 1, 51-89

on the lower part of page 58 the authors mention the manuscript

A. Machi, Growth functions and growth matrices for finitely generated groups. Unpublished manuscript, Univ. di Roma La Sapienza, 1986.

and explain an example due to Machi. Machi showed that the convergence of $b_{n+1}/b_n$ can fail for one generating set of ${mathbb Z}_2 star {mathbb Z}_3$ and hold for another. In particular, the limit does not always exist. The two generating sets are $lbrace s,trbrace$ and $lbrace s,strbrace$, where $s$ and $t$ are the natural generators with $s^2=t^3=e$.