A census of vertices by generations in regular tessellations of the plane (Q540084)

Summary: 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 \(p \geq 3\) and \(q \geq 3\) with \(1/p + 1/q \leq 1/2\), we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation.