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In crystallography and the theory of infinite vertex-transitive graphs, the coordination sequence of a vertex v {\displaystyle v} is an integer sequence that counts how many vertices are at each possible distance from v {\displaystyle v}. That is, it is a sequence
As an example, in a square grid, for each positive integer i {\displaystyle i} , there are 4 i {\displaystyle 4i} grid points that are i {\displaystyle i} steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence
The coordination sequences of many low-dimensional lattices and uniform tilings are known.
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