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A001931 Number of fixed 3-dimensional polycubes with n cells; lattice animals in the simple cubic lattice (6 nearest neighbors), face-connected cubes.
(Formerly M2996 N1213)
1, 3, 15, 86, 534, 3481, 23502, 162913, 1152870, 8294738, 60494549, 446205905, 3322769321, 24946773111, 188625900446, 1435074454755, 10977812452428, 84384157287999, 651459315795897 (list; graph; refs; listen; history; text; internal format)



This gives the number of polycubes up to translation (but not rotation or reflection). - Charles R Greathouse IV, Oct 08 2013


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

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Table of n, a(n) for n=1..19.

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Computing and Combinatorics, 12th Annual International Conference, COCOON 2006, Taipei, Taiwan, August 15-18, 2006, pp. 418-427.

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.

A Asinowski, G Barequet, Y Zheng, Polycubes with small perimeter defect, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, (2018).

Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (W├╝rzburg, Germany, 16-18 March 2020).

Andrew R. Conway, The design of efficient dynamic programming and transfer matrix enumeration algorithms, Journal of Physics A: Mathematical and Theoretical, 2 August 2017. For another version see arXiv, arXiv:1610.09806 [math.CO], 2016-2017.

Kevin L. Gong, Polyominoes Home Page

S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565.

S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Stat. Phys. 58 (5-6) (1990) 1095-1108, Table 1.

H. Redelmeier, Emails to N. J. A. Sloane, 1991


Cf. A000162, A001420, A151830, A151832, A151833, A151834, A151835.

Sequence in context: A326275 A093615 A191148 * A306524 A180677 A220875

Adjacent sequences:  A001928 A001929 A001930 * A001932 A001933 A001934




N. J. A. Sloane.


Edited by Arun Giridhar, Feb 14 2011

a(17) from Achim Flammenkamp, Feb 15 1999.

a(18) from the Aleksandrowicz and Barequet paper (N. J. A. Sloane, Jul 09 2009)

a(19) from Luther and Mertens by Gill Barequet, Jun 12 2011



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Last modified March 2 14:38 EST 2021. Contains 341751 sequences. (Running on oeis4.)