  By using the asymptotic theory of Pemantle and Wilson, exact asymptotic
expansions of the free energy of the monomer-dimer model on rectangular $n
\times \infty$ lattices in terms of dimer density are obtained for small values
of $n$, at both high and low dimer density limits. In the high dimer density
limit, the theoretical results confirm the dependence of the free energy on the
parity of $n$, a result obtained previously by computational methods. In the
low dimer density limit, the free energy on a cylinder $n \times \infty$
lattice strip has exactly the same first $n$ terms in the series expansion as
that of infinite $\infty \times \infty$ lattice.
