  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 $1/p + 1/q = 1/2$, these are tilings of the
Euclidean plane; for $1/p + 1/q < 1/2 $, 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/p + 1/q \le 1/2
$, we determine the rational generating function giving the number of vertices
in each generation.

Combinatorics