Enumeration of hybrid domino–lozenge tilings

We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every third diagonal drawn in. We also obtain a generalization of Douglasʼ theorem on the number of tilings of a family of regions of the square lattice with every second diagonal drawn in.