We extend the exact solutions of the Di Marzio-Rubin matrix method for the thermodynamic properties, including chain density, of a linear polymer molecule confined to walk on a lattice of finite size. Our extensions enable (a) the use of higher dimensions (explicit 2D and 3D lattices), (b) lattice boundaries of arbitrary shape, and (c) the flexibility to allow each monomer to have its own energy of attraction for each lattice site. In the case of the large chain limit, we demonstrate how periodic boundary conditions can also be employed to reduce computation time. Advantages to this method include easy definition of chemical and physical structure (or surface roughness) of the lattice and site-specific monomer-specific energetics, and straightforward relatively fast computations. We show the usefulness and ease of implementation of this extension by examining the effect of energy variation along the lattice walls of an infinite rectangular cylinder with the idea of studying the changes in properties caused by chemical inhomogeneities on the surface of the box. Herein, we look particularly at the polymer density profile as a function of temperature in the confined region for very long polymers. One particularly striking result is the shift in the critical condition for adsorption due to surface energy inhomogeneities and the length scale of the inhomogeneities; an observation that could have important implications for polymer chromatography. Our method should have applications to both copolymers and biopolymers of arbitrary molar mass.