Geodesic
A periodicity theorem for the octahedron recurrence
Journal of Algebraic Combinatorics, Tome 26 (2007) no. 1, pp. 1-26.

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Summary: The octahedron recurrence lives on a 3-dimensional lattice and is given by $f( x, y, t+1)=$( f( x+1, $y, t) f( x -1, y, t)+ f( x, y+1, t) f( x, y -1, t))/ f( x, y, t -1) f(x,y,t+1)=(f(x+1,y,t)f(x-1,y,t)+f(x,y+1,t)f(x,y-1,t))/f(x,y,t-1)$. In this paper, we investigate a variant of this recurrence which lives in a lattice contained in $[0, m] \times [0, n] \times \mathbb R$ [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.
Keywords: keywords octahedron recurrence, Laurent phenomenon, perfect matchings 
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Henriques, André. A periodicity theorem for the octahedron recurrence. Journal of Algebraic Combinatorics, Tome 26 (2007) no. 1, pp. 1-26. http://geodesic.mathdoc.fr/item/JAC_2007__26_1_a5/