Distance sequences and percolation thresholds in Archimedean tilings

Given a graph G, a vertex x of G and an integer n > 0, the circle Cn of center x and of radius n is the set of all the vertices at distance n from x and the circumference cn is the cardinality of Cn. The distance sequence of G and center x is the sequence (c0,c1,c2,…,cn,…). When G is vertex-transitive, we can talk about the distance sequence of G. In this note, we give formulae for calculating distance sequences in Archimedean tilings (which can be seen as vertex-transitive graphs) and we can see that, for these tilings, these sequences are linked with the percolation thresholds.