Compression Theorems for Periodic Tilings and Consequences

We consider a weighted square-and-domino tiling model obtained by assigning real number weights to the cells and boundaries of an n-board. An important special case apparently arises when these weights form periodic sequences. When the weights of an nm-tiling form sequences having period m, it is shown that such a tiling may be regarded as a meta-tiling of length n whose weights have period 1 except for the first cell (i.e., are constant). We term such a contraction of the period in going from the longer to the shorter tiling as “period compression.” It turns out that period compression allows one to provide combinatorial interpretations for certain identities involving continued fractions as well as for several identities involving Fibonacci and Lucas numbers (and their generalizations).