A periodicity theorem for the octahedron recurrence

Abstract

The octahedron recurrence lives on a 3-dimensional lattice and is given by \(f(x,y,t+1)=(f(x+1,y,t)f(x-1,y,t)+f(x,y+1,t)f(x,y-1,t))/f(x,y,t-1)\). In this paper, we investigate a variant of this recurrence which lives in a lattice contained in \([0,m] \times [0,n] \times \mathbb R\). 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.