The way particles pack together into jammed, mechanically stable structures, known as inherent structures, has implications for the thermodynamic, structural and dynamic properties of fluids, glasses and athermal granular systems. However, analytical results for packing problems in bulk, continuous systems are challenging and few exact results are known. Confining the system to narrow quasi-one-dimensional channels simplifies the packing problem and opens up the possibility of obtaining analytical results. It also introduces a new length scale that leads to the stabilization of packing environments not observed in bulk systems. This work develops a confined k-next nearest neighbour lattice model, the lattice version of hard spheres in the continuum, where the transfer matrix method is used to develop exact results for the thermodynamic properties of the fluid, as well as the inherent structures. The nature of the packings in confinement and the density distribution of inherent structures is examined as a function of the confining channel diameter.