  It is shown that the tight-binding approximation of the nonlinear
Schr\"odinger equation with a periodic linear potential and periodic in space
nonlinearity coefficient gives rise to a number of nonlinear lattices with
complex, both linear and nonlinear, neighbor interactions. The obtained
lattices present non-standard possibilities, among which we mention a
quasi-linear regime, where the pulse dynamics obeys essentially the linear
Schr{\"o}dinger equation. We analyze the properties of such models both in
connection with their modulational stability, as well as in regard to the
existence and stability of their localized solitary wave solutions.
