Two different methods of obtaining the partition function for a spin ½ open ended Ising chain with interactions between nearest and second nearest neighbour spins are presented. These are non-matrix methods, in that they do not require the diagonalization of a transfer matrix, and they provide a useful representation for studying the analytical structure of the partition function. The pair correlation functions are also derived, using the second of these methods. These correlation functions are utilized in a study of the ground state and the finite temperature spin configurations. An interesting application of these results to the problem of out-of-phase domain structure in binary alloys is discussed.