A periodicity theorem for the octahedron recurrence

The octahedron recurrence lives on a 3-dimensional lattice and is given by . In this paper, we investigate a variant of this recurrence which lives in a lattice contained in . Following Speyer, we give an explicit non-recursive formula for the values of this recurrence and use it to prove that it is periodic of period n+m. We then proceed to show various other hidden symmetries satisfied by this bounded octahedron recurrence. An earlier version of this work has circulated under the name “A coboundary category defined using the octahedron recurrence.”