Proper n-Cell Polycubes in n - 3 Dimensions

A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d - 1)-dimensional faces. Enumeration of polycubes, and, in particular, specific types of polycubes, as well as computing the asymptotic growth rate of polycubes, is a popular problem in combinatorics and discrete geometry. This is also an important tool in statistical physics for computations and analysis of percolation processes and collapse of branched polymers. A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. In this paper we prove that the number of polycubes of size n that are proper in n - 3 dimensions is 2(n-6)n(n-7)(n - 3)(12n(5) - 104n(4) + 360n(3) - 679n(2) + 1122n - 1560)/3.