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The number of squares in the Aztec diamond of order is , giving for , 2, OEIS A The number of domino tilings of an order Aztec diamond is , where is the triangular number Elkies et al.
Note that Wassermann appears to use an incorrect definition of the Aztec diamond that is actually equivalent to that of a centered square number.
Portions of this entry contributed by Gary Russell. Finding valid tilings of the Aztec diamond involves the solution of the underlying set-covering problem.
Two dominoes within D can be found to cover any boundary square within S, and four dominoes within D can be found to cover any non-boundary square within S.
With these definitions, the task of tiling the Aztec diamond may be reduced to a constraint satisfaction problem formulated as a binary integer program:.
This formulation can be solved with standard integer programming packages. Additional constraints can be constructed to force placement of particular dominoes, ensure a minimum number of horizontal or vertically-oriented dominoes are used, or generate distinct tilings.
An alternative approach is to apply Knuth's Algorithm X to enumerate valid tilings for the problem. There are many tools used throughout tiling projects, but two useful ones are GeoGebra and program created by Jim Propp , Greg Kuperberg , and David Wilson in SageMath to count the tilings of a shape.
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