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The first few are illustrated above. 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.
Each tile will cover exactly one black square. Vertical tiles where the top square covers a black square, is colored in one color, and the other vertical tiles in a second.
Similarly for horizontal tiles. Something that is very useful for counting tilings is looking at the non-intersecting paths through its corresponding directed graph.
If we define our movements through a tiling domino tiling to be. Then through any tiling we can have these paths from our sources to our sinks.
These movements are similar to Schröder paths. For example, consider an Aztec Diamond of order 2, and after drawing its directed graph we can label its sources and sinks.
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