A periodicity theorem for the octahedron recurrence

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.