  This article addresses the problem of enumerating the tilings of a plane by
lozenges, under the restriction that these tilings be doubly periodic.
Kasteleyn's Pfaffian method is applied to compute the generating function of
those permutations. The monomials of this function represent the different
types of tilings, grouping them according to the number of lozenges in each
orientation. We present an alternative approach to compute these types.
Finally, two additional classes of tilings are proposed as open enumeration
problems.
