2-adic behavior of numbers of domino tilings (Q1277790)

The number of domino tilings of a \(2n\) by \(2n\) square of points was proved by the reviewer and others to be \[ \prod^n_{i=1} \prod^n_{j=1} \left(4 \cos^2 {\pi i\over 2n+1} +4\cos^2 {\pi j \over 2n+1} \right). \] This number is known to be \(2^n[f(n)]^2\) where \(f(n)\) is an odd positive integer. ``Various results for \(f(n)\) are proved by elementary'' methods, including a simple difference equation in \(n\).