A new approach is developed for lattice density functional theory (LDFT) of symmetric dimers. Equations of equilibrium are derived for the complete set of configurations in the first three shells around the central dimer, and rules of truncation for higher shells are based on exact results from the mathematical theory of domino tilings. This provides exact limits, for both low and high densities. The new model predicts contributions of particular configurations which are in agreement with Monte Carlo simulations over the whole range of densities, including agreement with pocket Monte Carlo simulations at high densities.