Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.4 |

Department of Mathematics

The Technion–Israel Institute of Technology

Haifa 32000

Israel

Gill Barequet

Department of Computer Science

The Technion–Israel Institute of Technology

Haifa 32000

Israel

Ronnie Barequet

Department of Computer Science

Tel Aviv University

Tel Aviv 69978

Israel

Günter Rote

Institut für Informatik

Freie Universität Berlin

Takustraße 9

D-14195 Berlin

Germany

**Abstract:**

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)
(12*n*^{5} - 104*n*^{4} + 360*n*^{3}
- 679*n*^{2} + 1122*n* - 1560) / 3.

(Concerned with sequences A127670 A171860 A191092.)

Received June 6 2012;
revised version received October 1 2012.
Published in *Journal of Integer Sequences*, October 2 2012.

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