We prove that the number of monomer-dimer tilings of an n×n square grid, with m<n monomers in which no four tiles meet at any point is m2m+(m+1)2m+1, when m and n have the same parity. In addition, we present a new proof of the result that there are n2n−1 such tilings with n monomers, which divides the tilings into n classes of size 2n−1. The sum of these tilings over all monomer counts has the closed form 2n−1(3n−4)+2 and, curiously, this is equal to the sum of the squares of all parts in all compositions of n. We also describe two algorithms and a Gray code ordering for generating the n2n−1 tilings with n monomers, which are both based on our new proof.