A191092 Number of n-cell polycubes that are proper in n-3 dimensions.

0, 1, 61, 2836, 129288, 6160640, 313921008, 17239040000, 1021644763392, 65244849242112, 4477975127425280, 329252714454974464, 25850313756000000000, 2160223055912342913024, 191558954408834121740288, 17973564914103712921681920

Offset

3,3

References

A. Asinowski, G. Barequet, R. Barequet, and G. Rote, Proper n-cell polycubes in n-3 dimensions, Proc. 17th Ann. Int. Computing and Combinatorics Conference, Dallas, TX, Lecture Notes in Computer Science, 6842, Springer-Verlag, 180-191, August 2011.

Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf

G. Barequet, M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.

R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275.

Links

Vincenzo Librandi, Table of n, a(n) for n = 3..100

A. Asinowski, G. Barequet, R. Barequet, and G. Rote, Proper n-Cell Polycubes in n-3 Dimensions, J. Int. Seq. 15 (2012) #12.8.4.

M. Shalah, Formulae and growth rates of animals on cubical and triangular lattices, PhD Thesis, Israel Inst. Techn. (2017)

Formula

a(n) = 2^(n-6)*n^(n-7)*(n-3)*(12*n^5 - 104*n^4 + 360*n^3 - 679*n^2 + 1122*n - 1560)/3.

Mathematica

a[n_]:=2^(n-6)*n^(n-7)*(n-3)*(12*n^5 - 104*n^4 + 360*n^3 - 679*n^2 + 1122*n - 1560)/3 ; Array[a, 40, 3] (* Stefano Spezia, Sep 09 2018 *)

Prog

(PARI) a(n)=2^(n-6)*n^(n-7)*(n-3)*(12*n^5-104*n^4+360*n^3-679*n^2+1122*n-1560)/3
(Magma) [2^(n-6)*n^(n-7)*(n-3)*(12*n^5-104*n^4+360*n^3-679*n^2+1122*n-1560)/3: n in [3..40]]; // Vincenzo Librandi, May 26 2011

Crossrefs

Cf. A127670, A171860.

Diagonal 3 of A195739.

Sequence in context: A224441 A078962 A000508 * A234028 A135647 A269025

Adjacent sequences: A191089 A191090 A191091 * A191093 A191094 A191095

Keyword

nonn,easy

Author

Gill Barequet, May 25 2011

Status

approved