In
crystallography and the theory of infinite
vertex-transitive graphs, the coordination sequence of a vertex is an
integer sequence that counts how many vertices are at each possible distance from . That is, it is a sequence
where each is the number of vertices that are steps away from . If the graph is vertex-transitive, then the sequence is an
invariant of the graph that does not depend on the specific choice of . Coordination sequences can also be defined for
sphere packings, by using either the
contact graph of the spheres or the
Delaunay triangulation of their centers, but these two choices may give rise to different sequences.[1][2]
As an example, in a
square grid, for each positive integer , there are grid points that are steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence
in which, except for the initial value of one, each number is a multiple of four.[3]