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
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)
Programs
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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
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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 *)
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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
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.