We consider the equilibrium statistical mechanics of a classical one-dimensional simple fluid, with nearest-neighbor interactions, and in an arbitrary external potential. The external potential is eliminated to yield relations between the truncated partition functions and the one-body density. These relations are solved for pure cores and for sticky cores, resulting in each case in both an explicit potential density relation and grand potential density functional. Both models maintain finite-range direct correlations and have grand potentials expressible in terms of simple linear density transforms.