Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

A d-dimensional polycube is a face-connected set of cells on Z(d). Let A(d) (n) denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and lambda(d) denote their growth constant lim(n ->infinity) A(d)(n+1)/A(d)(n). We revisit and extend the method for the best known upper bound on A(2)(n). Our contributions include the following. 1. We prove that lambda(2) <= 4.5252; 2. We prove that lambda(d) <= (2d - 2)e + o(1) for d >= 2 (already improving significantly the known upper bound on lambda(3) to 9.8073); and 3. We implement an iterative process in three dimensions, improving further the upper bound on lambda(3) to 9.3835.