A001931 -id:A001931 - cp's OEIS frontend

A151835 Number of fixed 9-dimensional polycubes with n cells.

1, 9, 153, 3309, 81837, 2205489, 63113061, 1887993993, 58441956579, 1858846428437, 60445700665383, 2001985304489169, 67341781440810531, 2295424989986481345

Offset: 1

N. J. A. Sloane, Jul 12 2009

a(1)-a(10) can be computed by formulas in Barequet et al. (2010). Luther and Mertens confirm these values (and add two more) by direct counting.

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
Gill Barequet, Gil Ben-Shachar, and Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes; in: Handbook of Discrete and Computational Geometry, Chapman and Hall/CRC, 2017. (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.
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275.
S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), P09026.
Stephan Mertens, Lattice Animals

Cf. A001931, A151830, A151831, A151832, A151833, A151834.

a(5)-a(12) from Luther and Mertens by Gill Barequet, Jun 12 2011

a(13)-a(14) from Mertens added by Andrey Zabolotskiy, Jan 29 2023

A366335 Number of fixed (4,2)-polyominoids with n cells.

Original entry on oeis.org

6, 60, 916, 16698, 336210, 7218768, 162185112, 3769221330, 89924613880

Offset: 1

Pontus von Brömssen, Oct 07 2023

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes with integer coordinates in D-dimensional space, where two cubes are connected if they share a (d-1)-dimensional facet. For example, (3,2)-polyominoids are normal polyominoids (A075678), (D,D)-polyominoids are D-dimensional polyominoes (A001168, A001931, A151830, ...), and (D,1)-polyominoids are polysticks in D dimensions (A096267, A365560, A365562, ...).

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A366334 (free).
46th row of A366767.
Fixed (D,d)-polyominoids:
  D\d|    1       2       3       4
  ---+--------------------------------
   1 | A000012
   2 | A096267 A001168
   3 | A365560 A075678 A001931
   4 | A365562 A366335 A366337 A151830

a(7)-a(9) from John Mason, Jul 05 2025

A007743 Number of achiral polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4} (or polycubes).

Original entry on oeis.org

1, 1, 2, 6, 17, 58, 191, 700, 2515, 9623, 36552, 143761, 564443, 2259905, 9057278, 36705846, 149046429, 609246350, 2495727647, 10267016450, 42322763940, 174974139365

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

A000162 but with both copies of each mirror-image deleted.

An achiral polyomino is identical to its reflection. Many of these achiral polyominoes do not have a plane of symmetry. For example, the hexomino with cell centers (0,0,0), (0,0,1), (0,1,1), (1,1,1), (1,2,1), and (1,2,2) has a center of symmetry at (1/2,1,1) but no plane of symmetry. The decomino with cell centers (0,0,0), (0,0,1), (0,1,1), (0,2,1), (0,2,2), (1,0,2), (1,1,2), (1,1,1), (1,1,0), and (1,2,0) has no plane or center of symmetry. - Robert A. Russell, Mar 21 2024

G. Thimm, Emails to N. J. A. Sloane, Sep. 1994

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.
Cf. A000162 (oriented), A038119 (unoriented), A371397 (chiral), A001931 (fixed).
Component symmetries: A376969, A376970, A376971, A376972, A376973, A376974, A376977, A376978, A376979, A376980, A376981, A376983, A376984, A376985, A376986, A376987, A376988, A376989, A376990, A376991, A377129, A377130.

a(n) = A000162(n) - 2*A371397(n) = A038119(n) - A371397(n). - Robert A. Russell, Mar 21 2024

a(13)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
Changed "symmetric" to "mirror-symmetric" in the title by George Sicherman, Feb 21 2018
Changed "mirror-symmetric" to "achiral" in the title to ensure that a plane of symmetry is not required. - Robert A. Russell, Mar 21 2024
a(17)-a(22) from John Mason, Sep 19 2024

A229914 Number of pyramid polycubes of a given volume in dimension 3.

Original entry on oeis.org

1, 3, 7, 16, 33, 63, 117, 202, 344, 566, 908, 1419, 2206, 3334, 4988, 7378, 10778, 15535, 22281, 31547, 44405, 62011, 85939, 118281, 162136, 220494, 298531, 402163, 539181, 719301, 956287, 1265022, 1667973, 2190934, 2867470, 3739797, 4864163, 6303461, 8146863, 10499087, 13493267, 17293169, 22111954

Offset: 1

Matthieu Deneufchâtel, Oct 03 2013

nonn

A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell of coordinate (0,b,c) belonging to the first plateau is such that b , c >= 0.

If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau.

C. Carré, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4; also hal-00905889, version 1.

A001931 is an upper bound.

The generating function for the numbers of pyramids of height h and volumes v_1 , ... v_h is (n_1-n_2+1) *(n_2-n_3+1) *... *(n_{h-1}-n_h+1) *(x_1^{n_1} * ... x_h^{n_h}) / ((1-x_1^{n_1}) *(1-x_1^{n_1}*x_2^{n_2}) *... *(1-x_1^{n_1}*x_2^{n_2}*...x_h^{n_h})).

This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>=, ... >= n_h>=1 and then over h.

A371397 Number of chiral pairs of polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}.

Original entry on oeis.org

0, 0, 0, 1, 6, 54, 416, 3111, 22898, 168460, 1242985, 9227333, 68949103, 518618196, 3925228596, 29879207817, 228630283775, 1757699977107, 13570824097968, 105182547181534, 818093724437992, 6383353614308209

Offset: 1

Robert A. Russell, Mar 21 2024

Also called polycubes. Each member of a chiral pair is a reflection but not a rotation of the other.

			Polyominoes with cell centers at (0,0,0), (0,0,1), (0,1,1), (1,1,1) and (0,0,0), (0,1,0), (0,1,1), (1,1,1) are a chiral pair.

Cf. A000162 (oriented), A038119 (unoriented), A007743 (achiral), A001931 (fixed).

Component symmetries: A376964, A376965, A376966, A376967, A376968, A376975, A376976, A376982, A377127, A377128, A377131.

a(n) = A000162(n) - A038119(n) = (A000162(n) - A007743(n))/2 = A038119(n) - A007743(n).

a(17)-a(22) from John Mason, Sep 19 2024

A385291 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional polyominoes of size k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 19, 15, 4, 1, 1, 63, 86, 28, 5, 1, 1, 216, 534, 234, 45, 6, 1, 1, 760, 3481, 2162, 495, 66, 7, 1, 1, 2725, 23502, 21272, 6095, 901, 91, 8, 1, 1, 9910, 162913, 218740, 80617, 13881, 1484, 120, 9, 1, 1, 36446, 1152870, 2323730, 1121075, 231008, 27468, 2276, 153, 10, 1

Offset: 1

John Mason, Jun 24 2025

			The top corner of the array (size on horizontal axis, dimensions on vertical):
          1: 1  1    1     1       1         1           1
(A001168) 2: 1  2    6    19      63       216         760
(A001931) 3: 1  3   15    86     534      3481       23502
(A151830) 4: 1  4   28   234    2162     21272      218740
(A151831) 5: 1  5   45   495    6095     80617     1121075
(A151832) 6: 1  6   66   901   13881    231008     4057660
(A151833) 7: 1  7   91  1484   27468    551313    11710328
(A151834) 8: 1  8  120  2276   49204   1156688    28831384
(A151835) 9: 1  9  153  3309   81837   2205489    63113061
         10: 1 10  190  4615  128515   3906184   126210640
         11: 1 11  231  6226  192786   6524265   234919234
         12: 1 12  276  8174  278598  10389160   412504236
         13: 1 13  325 10491  390299  15901145   690185431
         14: 1 14  378 13209  532637  23538256  1108774772
         15: 1 15  435 16360  710760  33863201  1720467820
         16: 1 16  496 19976  930216  47530272  2590788848
         17: 1 17  561 24089 1196953  65292257  3800689609
         18: 1 18  630 28731 1517319  88007352  5448801768
         19: 1 19  703 33934 1898062 116646073  7653842998
         20: 1 20  780 39730 2346330 152298168 10557176740
         21: 1 21  861 46151 2869671 196179529 14325525627
         22: 1 22  946 53229 3476033 249639104 19153838572
         23: 1 23 1035 60996 4173764 314165809 25268311520
         24: 1 24 1128 69484 4971612 391395440 32929561864

John Mason, Table of n, a(n) for n = 1..162

Cf. A000384 (column k=3), A195739.

Rows: A000012 (n=1), A001168 (n=2), A001931 (n=3), A151830 (n=4), A151831 (n=5), A151832 (n=6), A151833 (n=7), A151834 (n=8), A151835 (n=9).

A(n,k) = Sum_{d=0..n} binomial(n,d)*A195739(k,d) (with A195739(k,d) = 0 for k <= d). - Pontus von Brömssen, Jun 28 2025

a(56)-a(66) from Pontus von Brömssen, Jun 28 2025

A006763 Number of fixed properly-3-dimensional polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 32, 348, 2836, 21225, 154741, 1123143, 8185403, 60088748, 444688325, 3317057654, 24925158492, 188543716451, 1434760675947, 10976610064899, 84379534826376, 651441493579872

Offset: 1

N. J. A. Sloane

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

W. F. Lunnon, Counting multidimensional polyominoes. Computer Journal 18 (1975), no. 4, pp. 366-367.

A column of A195739.

A001931 = Cases[Import["https://oeis.org/A001931/b001931.txt", "Table"], {, }][[All, 2]];
A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 1, 0, A001931[[n]] - 3 (A001168[[n]] - 1)];
a /@ Range[1, 19] (* Jean-François Alcover, Sep 22 2019 *)

a(n) = A001931(n) - 3 * (A001168(n) - 1) for n > 1. - Sean A. Irvine, Jul 27 2017

More terms from Jean-François Alcover, Sep 22 2019

A113174 Number of fixed 3D piled polyominoes: polycubes with fixed orientation, with no cubes "sitting on air".

Original entry on oeis.org

1, 3, 11, 44, 184, 792, 3484, 15592, 70745, 324561, 1502511, 7007929, 32892778, 155221536, 735915652, 3503270920, 16737092549, 80218277681, 385574074383, 1858059853316, 8974761939239, 43441619693731, 210682920968681

Offset: 1

Franklin T. Adams-Watters, Oct 15 2005

nonn

			For n = 4, there are 4 orientations of the angled tricube excluded: those which set it on a point; this leaves 8 orientations of the angled tricube and 3 of the straight tricube.

Cf. A001168, A001931 (fixed polycubes).

a(n) = sum_{m=1}^n A001168(m)*C(n-1, m-1). If both sequences are shifted left, binomial transform of A001168.

Corrected by Franklin T. Adams-Watters, Oct 25 2006

A363090 Number of 3-dimensional directed animals of size n.

Original entry on oeis.org

1, 3, 12, 52, 237, 1113, 5339, 26011, 128247, 638346, 3201967, 16164384, 82044151, 418352107, 2141761669, 11003117220

Offset: 1

Joerg Arndt and Márk Péter Légrádi, May 19 2023

Fixed 3-dimensional fixed polycubes with neighborhood vectors [1, 0, 0], [0, 1, 0], and [0, 0, 1].
A005773 is the 2-dimensional equivalent: fixed polyominoes with neighborhood vectors [1, 0] and [0, 1].
Using neighborhood vectors [+-1, 0, 0], [0, +-1, 0], and [0, 0, +-1] gives A001931.

Cf. A005773.

a(14)-a(16) from Johann Peters, Nov 21 2024

cp's OEIS Frontend

A151835 Number of fixed 9-dimensional polycubes with n cells.

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A366335 Number of fixed (4,2)-polyominoids with n cells.

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A007743 Number of achiral polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4} (or polycubes).

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A229914 Number of pyramid polycubes of a given volume in dimension 3.

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A371397 Number of chiral pairs of polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}.

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A385291 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional polyominoes of size k.

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A006763 Number of fixed properly-3-dimensional polyominoes with n cells.

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A113174 Number of fixed 3D piled polyominoes: polycubes with fixed orientation, with no cubes "sitting on air".

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A363090 Number of 3-dimensional directed animals of size n.

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