A Census of Vertices by Generations in Regular Tessellations of the Plane

We consider regular tessellations of the plane as infinite graphs in which $q$ edges and $q$ faces meet at each vertex, and in which $p$ edges and $p$ vertices surround each face. For $1\mathrm{/}p+1\mathrm{/}q=1\mathrm{/}2$, these are tilings of the Euclidean plane; for $1\mathrm{/}p+1\mathrm{/}q<1\mathrm{/}2$, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all $p\ge 3$ and $q\ge 3$ with $1\mathrm{/}p+1\mathrm{/}q\le 1\mathrm{/}2$, we determine the rational generating function giving the number of vertices in each generation.