A coloring-book approach to finding coordination sequences.
A coloring-book approach to finding coordination sequences.
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寻找协调序列的着色书方法。
DOI：
10.1107/s2053273318014481
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
N. J. A. Sloane
中科院分区：
文献类型：
--
作者：
C. Goodman;N. J. A. Sloane
描述了一种基本的方法，用于找到协调序列的平铺，着色的基础图的基础上。第一个应用程序是两种顶点（四价和三价）在开罗（或双-32.4.3.4）平铺。令人惊讶的是，四价顶点的配位顺序是1，4，8，12，16，...，与熟悉的正方形（或44）镶嵌中的顶点相同。作者认为这样一个简单的事实应该有一个简单的证明，本文就是这样的结果。该方法还用于获得32.4.3.4、3.4.6.4、4.82、3.122和34.6均匀平铺以及snub-632平铺的协调序列。在某些情况下，这些结果为以前提出的公式提供了证明。
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, 3.122 and 34.6 uniform tilings, and the snub-632 tiling. In several cases the results provide proofs for previously conjectured formulas.