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We study the ground state properties of a one-dimensional Ising chain with a nearest-neighbor ferromagnetic interactionJ1, and akth neighboranti-ferromagnetic interactionJk. WhenJk/J1=-1/k, there exists a highly degenerate ground state with a residual entropy per spin. For the finite chain with free boundary conditions, we calculate the degeneracy of this state exactly, and find that it is proportional to the (N+k-1)th term in a generalized Fibonacci sequence defined by,FN(k)=FN-1(k)+FN-k(k). In addition, we show that this one-dimensional model is closely related to the following problems: (a) a fully frustrated two-dimensional Ising system with a periodic arrangement of nearest-neighbor ferro- and antiferromagnetic bonds, (b) close-packing of dimers on a ladder, a 2×8 strip of the square lattice, and (c) “directed” self-avoiding walks on finite lattice strips.

One-dimensional Ising chain with competing interactions: Exact results and connection with other statistical models