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] × [0,n] × 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. © 2006 Springer Science+Business Media, LLC.