When I saw the recent question [Kasteleyn's formula for domino tilings generalized?][1], I had the idea of generalizing it by [introducing an additional variable][2]. I didn't manage to solve the generalized question yet, but working on it, it suddenly occurred to me a couple of days later that the additional variable provided in fact [an easy answer][3] of the initial question. 


  [1]: http://mathoverflow.net/questions/258386/kasteleyns-formula-for-domino-tilings-generalized?noredirect=1&lq=1
  [2]: http://mathoverflow.net/questions/258530/generalizing-kasteleyns-formula-even-more?noredirect=1&lq=1
  [3]: http://mathoverflow.net/questions/258386/kasteleyns-formula-for-domino-tilings-generalized/259530#259530