Let $G_N$ be the group generated by elements $a_1,\ldots,a_N$ subject to the relations $a_i^2=1$ and $(a_ia_j)^3=1$. The growth function of $G_N$ is then $$f_N(t)=\frac{1+2t+2t^2+t^3}{1-Mt-Mt^2+\frac{M(M+1)}{2}t^3}$$ where $M=N-2$ (or at least, this seems to be the correct generalization of the $N=4,5$ known formulae, A154638 and A162740 @ OEIS). Question now: is the random walk function $$g_N(t)=\sum_{k=0}^\infty Card(i_1,\ldots,i_k|a_{i_1}\ldots a_{i_k}=1)\cdot t^k$$ known? Maybe related to $f_N$? I'd be actually interested in the $N\to\infty$ behavior of $g_N$.