  An elementary method is described for finding the coordination sequences for
a tiling, based on coloring the underlying graph. We illustrate the method by
first applying it to the two kinds of vertices (tetravalent and trivalent) in
the Cairo (or dual-3^2.4.3.4) tiling. The coordination sequence for a
tetravalent vertex turns out, surprisingly, to be 1, 4, 8 ,12, 16, ..., the
same as for a vertex in the familiar square (or 4^4) tiling. We thought that
such a simple fact should have a simple proof, and this article is the result.
We also apply the method to obtain coordination sequences for the 3^2.4.3.4,
3.4.6.4, 4.8^2, 3.12^2, and 3^4.6 uniform tilings, as well as the snub-632 and
bew tilings. In several cases the results provide proofs for previously
conjectured formulas.
