Formulae and Growth Rates of High-Dimensional Polycubes

A d-dimensional polycube is a facet-connected set of cubes in d dimensions. Fixed polycubes are considered distinct if they differ in shape or orientation. A proper d-D polycube spans all d dimensions. In this paper we prove some formulae for fixed (proper and improper) polycubes, show that the growth-rate limit of the number of polycubes in d dimensions is 2ed−o(d), and estimate it at (2d−3)e+O(1/d).