$2$-adic Behavior of Numbers of Domino Tilings

Abstract

We study the $2$-adic behavior of the number of domino tilings of a $2n \times 2n$ square as $n$ varies. It was previously known that this number was of the form $2^nf(n)^2$, where $f(n)$ is an odd, positive integer. We show that the function $f$ is uniformly continuous under the $2$-adic metric, and thus extends to a function on all of $Z$. The extension satisfies the functional equation $f(-1-n) = \pm f(n)$, where the sign is positive iff $n \equiv 0,3 \pmod{4}$.