Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Abstract

A d-dimensional polycube is a face-connected set of cells on Zd. Let Ad(n) denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and λd denote their growth constant limn→∞Ad(n+1)Ad(n). We revisit and extend the method for the best known upper bound on A2(n). Our contributions: We (1) prove that λ2≤ 4.5252 ; (2) prove that λd≤ (2 d- 2 ) e+ o(1 ) for d≥ 2 (already improving significantly the upper bound on λ3 to 9.8073); and (3) implement an iterative process in 3D, improving further the upper bound on λ3 to 9.3835.