  The classical monomer-dimer model in two-dimensional lattices has been shown
to belong to the \emph{``#P-complete''} class, which indicates the problem is
computationally ``intractable''. We use exact computational method to
investigate the number of ways to arrange dimers on $m \times n$
two-dimensional rectangular lattice strips with fixed dimer density $\rho$. For
any dimer density $0 < \rho < 1$, we find a logarithmic correction term in the
finite-size correction of the free energy per lattice site. The coefficient of
the logarithmic correction term is exactly -1/2. This logarithmic correction
term is explained by the newly developed asymptotic theory of Pemantle and
Wilson. The sequence of the free energy of lattice strips with cylinder
boundary condition converges so fast that very accurate free energy $f_2(\rho)$
for large lattices can be obtained. For example, for a half-filled lattice,
$f_2(1/2) = 0.633195588930$, while $f_2(1/4) = 0.4413453753046$ and $f_2(3/4) =
0.64039026$. For $\rho < 0.65$, $f_2(\rho)$ is accurate at least to 10 decimal
digits. The function $f_2(\rho)$ reaches the maximum value $f_2(\rho^*) =
0.662798972834$ at $\rho^* = 0.6381231$, with 11 correct digits. This is also
the \md constant for two-dimensional rectangular lattices. The asymptotic
expressions of free energy near close packing are investigated for finite and
infinite lattice widths. For lattices with finite width, dependence on the
parity of the lattice width is found. For infinite lattices, the data support
the functional form obtained previously through series expansions.
