Connection with Aztec Diamond! Something interest, but only for (2n)x(2n) squares:

Consider the 2n-by-2n matrix View the MathML source with mi,j=1 for i,j satisfying |2i−2n−1|+|2j−2n−1|less-than-or-equals, slant2n and mi,j=0 for all other i,j, consisting of a central diamond of 1's surrounded by 0's. When ngreater-or-equal, slanted4, the λ-determinant of the matrix M (as introduced by Robbins and Rumsey [Adv. Math. 62 (1986) 169–184]) is not well defined. However, if we replace the 0's by t's, we get a matrix whose λ-determinant is well defined and is a polynomial in λ and t. The limit of this polynomial as t→0 is a polynomial in λ whose value at λ=1 is the number of domino-tilings of a 2n-by-2n square.