  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 Douglas'
Theorem on the number of tilings of a family of regions of the square lattice
with every second diagonal drawn in.
