  We study a simple heteropolymer model containing sequence-independent local
interactions on both square and triangular lattices. Sticking to a two-letter
code, we investigate the model for varying strength $\kappa$ of the local
interactions; $\kappa=0$ corresponds to the well-known HP model [K.F. Lau and
K.A. Dill, Macromolecules 22, 3986 (1989)]. By exhaustive enumerations for
short chains, we obtain all structures which act as a unique and pronounced
energy minimum for at least one sequence. We find that the number of such
designable structures depends strongly on $\kappa$. Also, we find that the
number of designable structures can differ widely for the two lattices at a
given $\kappa$. This is the case, for example, at $\kappa=0$, which implies
that the HP model exhibits different behavior on the two lattices. Our findings
clearly show that sequence-independent local properties of the chains can play
an important role in the formation of unique minimum energy structures.
