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We consider the dimer model on the rectangular $2M \times 2N$ lattice withfree boundary conditions. We derive exact expressions for the coefficients inthe asymptotic expansion of the free energy in terms of the elliptic thetafunctions ($\theta_2, \theta_3, \theta_4$) and the elliptic integral of secondkind ($E$), up to 22nd order. Surprisingly we find that ratio of thecoefficients $f_p$ in the free energy expansion for strip($f_p^\mathrm{strip}$) and square ($f_p^\mathrm{sq}$) geometries$r_p={f_p^\mathrm{strip}}/{f_p^\mathrm{sq}}$ in the limit of large $p$ tends to$1/2$. Furthermore, we predict that the ratio of the coefficients $f_p$ in thefree energy expansion for rectangular ($f_p(\rho)$) for aspect ratio $\rho > 1$to the coefficients of the free energy for square geometries, multiplied by$\rho^{-p-1}$, that is $r_p=\rho^{-p-1} {f_p(\rho)}/{f_p^\mathrm{sq}}$, is alsoequal to $1/2$ in the limit of $p \to \infty$. We find that the cornercontribution to the free energy for the dimer model on rectangular $2M \times2N$ lattices with free boundary conditions is equal to zero and explain thatresult in the framework of conformal field theory, in which the central chargeof the considering model is $c=-2$. We also derive a simple exact expressionfor the free energy of open strips of arbitrary width.

Exact finite-size corrections in the dimer model on a planar square lattice