A model of monomer adsorption on infinitely long, finite-width M equilateral triangular lattices with nonperiodic boundaries is presented. The study includes adsorbate-adsorbate first- and second-neighbor interactions with results obtained for repulsive first neighbors. The matrix method and numerical algorithms presented here allow determination of the occupational characteristics of the adsorption crystallization phases, which fit exact analytic expressions in the width M of the lattice. The limit as M approaches infinity provides the complete energy phase diagram for the infinite two-dimensional surface and recovers the results obtained by different methods that were often applied only in restricted energy regions of the phase diagram. The ordered phases are (2 x 1), (2 x 2), (3 x 1), (root 3 x root 3) R30 degrees, and the complementary phases of (2 x 2) and (root 3 x root 3) R30 degrees. Comparison is made with other theoretical studies and with experimental observations on adsorption systems consistent with the limitations of the model. In some cases, comparison with experimental data yields bounds on the interaction energies between adsorbates. On the basis of the model, suggestions are made on the manner in which to conduct relatively low temperature experiments to allow determination of most, if not all, of the interaction energies from the knowledge of the sequences of phases and the conditions prevailing at the transitions between phases.