  The inverse Kasteleyn matrix of a bipartite graph holds much information
about the perfect matchings of the system such as local statistics which can be
used to compute local and global asymptotics. In this paper, we consider three
different weightings of domino tilings of the Aztec diamond and show using
recurrence relations, we can compute the inverse Kasteleyn matrix. These
weights are the one-periodic weighting where the horizontal edges have one
weight and the vertical edges have another weight, the q^{vol} weighting which
corresponds to multiplying the product of tile weights by q if we add a `box'
to the height function and the two-periodic weighting which exhibits a flat
region with defects in the center.
