A coloring-book approach to finding coordination sequences
C. Goodman-Straussa, N. Sloaneb
- aDepartment of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA
- bThe OEIS Foundation Inc., 11 So. Adelaide Ave., Highland Park, NJ 08904, USA
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Abstract
An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-32.4.3.4) 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 44) tiling. The authors thought that such a simple fact should have a simple proof, and this article is the result. The method is also used to obtain coordination sequences for the 32.4.3.4, 3.4.6.4, 4.82, 3.122and 34.6 uniform tilings, and the snub-632 tiling. In several cases the results provide proofs for previously conjectured formulas.
TL;DRAn elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph, which turns out to be the same as for a vertex in the familiar square tiling. (Semantic Scholar)