A001931 -id:A001931 - cp's OEIS frontend

A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

1, 1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663, 228779330204, 1758309223457, 13573319825615, 105192814197984, 818136047201932, 6383528588447574

Offset: 1

N. J. A. Sloane

a(1)-a(12) computed by Achim Flammenkamp.
A000162 but with one copy of each mirror-image deleted.
From R. J. Mathar, Mar 19 2018: (Start)
We can split the numbers into an irregular table which lists in row n how many configurations have c contacts for c >= 0:
  1;
  0 1;
  0 0 2;
  0 0 0 6 1;
  0 0 0 0 21 2;
  0 0 0 0 0 91 19 2;
  0 0 0 0 0 0 484 110 12 1;
  0 0 0 0 0 0 0 2817 852 129 12 0 1;
  0 0 0 0 0 0 0 0 17788 6321 1166 132 5 1;
Row lengths are 1+A007818(n). Row sums are a(n).
(End)
Number of unoriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For unoriented polyominoes, chiral pairs are counted as one.- Robert A. Russell, Mar 21 2024

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
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. [See https://books.google.nl/books?id=ja7iBQAAQBAJ&pg=PA101]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Achim Flammenkamp, Home page.
Kevin L. Gong, Polyominoes Home Page.
John Mason, Counting free polycubes.
John Mason, Coordinate sets of examples of polycube symmetry (version 2)

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.
32nd row of A366766.
Cf. A000162 (oriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).
Cf. for each symmetry: A376964, A376965, A376966, A376967, A376968, A376969, A376970, A376972, A376973, A376974, A376975, A376976, A376977, A376978, A376979, A376980, A376981, A376982, A376983, A377127, A376984, A376985, A376986, A376987, A376988, A376989, A377128, A376990, A376991, A377129, A377130, A377131, A376971

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000162 = A@000162;
A007743 = A@007743;
a[n_] := (A007743[[n]] + A000162[[n]])/2;
a /@ Range[16] (* Jean-François Alcover, Jan 16 2020 *)

a(n) = A000162(n) - A371397(n) = A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Jan 02 2002
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
More terms from John Mason, Sep 19 2024

A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells.

Original entry on oeis.org

1, 1, 2, 8, 29, 166, 1023, 6922, 48311, 346543, 2522522, 18598427, 138462649, 1039496297, 7859514470, 59795121480, 457409613979, 3516009200564, 27144143923583, 210375361379518, 1636229771639924, 12766882202755783

Offset: 1

N. J. A. Sloane and J. H. Conway

Here two polycubes that differ by reflection are considered different. - Joerg Arndt, Apr 26 2023

Number of oriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Mar 21 2024

			Table showing total number and numbers with each group order.
-------------------------------------------------------------
The last 7 columns form sequences A066453, A066454, A066273, A066281, A066283, A066287, A066288.
.n ...A000162 ..group:.1.....2...3...4.6.8.24
.1 .........1..........0.....0...0...0.0.0..1
.2 .........1..........0.....0...0...0.0.1..0
.3 .........2..........0.....1...0...0.0.1..0
.4 .........8..........1.....4...1...0.0.2..0
.5 ........29.........17....10...0...0.0.2..0
.6 .......166........127....34...0...3.1.1..0
.7 ......1023........941....71...4...5.0.1..1
.8 ......6922.......6662...246...0..11.0.2..1
.9 .....48311......47771...522...3..11.0.4..0
10 ....346543.....344708..1783..24..24.2.2..0
11 ...2522522....2518713..3765...4..35.0.5..0
12 ..18598427...18585455.12858..18..84.5.7..0
13 .138462649..138434899.27496.151..92.2.8..1
14 1039496297.1039401564.94525..25.174.4.5..0

C. J. Bouwkamp, personal communication.
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.
W. F. Lunnon, personal communication.
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).

Nina Bohlmann and Ralf Benölken, Complex Tasks: Potentials and Pitfalls, Mathematics (2020) Vol. 8, No. 10, 1780.
C. J. Bouwkamp, A. J. W. Duijvestijn, & N. J. A. Sloane, Correspondence, 1971.
A. Clarke, Polycubes.
A. Clarke, The 8 tetracubes.
Stanley Dodds, C# program for this sequence.
Kevin L. Gong, Polyominoes Home Page.
M. Keller, Counting polyforms.
David A. Klarner, Some results concerning polyominoes, Fibonacci Quarterly 3 (1965), 9-20.
Jeffrey R. Long and R. H. Holm, Enumeration and structural classification of clusters derived from parent solids ..., J. Amer. Chem. Soc., 116 (1994), 9987-10002.
John Mason, Coordinate sets of examples of polycube symmetry (version 2)
Phillip Thompson, Rust port of Stanley Dodds's algorithm.
Eric Weisstein's World of Mathematics, Polycube.

Cf. A001931, A066453.

Cf. A038119 (unoriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).

a(n) = A066453 + A066454 + A066273 + A066281 + A066283 + A066287 + A066288.
a(n) = 2*A038119 - A007743.
a(n) = A000105 + A006759.
a(n) = A038119(n) + A371397(n) = 2*A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

The old value for a(11), 2522572, was corrected by Achim Flammenkamp to 2522522, Feb 15 1999.
a(13)-a(14) from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27 2001
a(15)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(17)-a(20) from Stanley Dodds, Dec 11 2023
a(21)-a(22) (using Dodds's algorithm) from Phillip Thompson, Feb 07 2024

A366767 Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 2, 4, 2, 0, 1, 0, 2, 12, 6, 1, 0, 1, 0, 2, 38, 22, 0, 1, 0, 1, 0, 2, 126, 88, 0, 2, 1, 0, 1, 0, 2, 432, 372, 0, 6, 2, 1, 0, 1, 0, 2, 1520, 1628, 0, 19, 6, 4, 3, 0, 1, 0, 2, 5450, 7312, 0, 63, 19, 20, 0, 3

Offset: 1

Pontus von Brömssen, Oct 22 2023

See A366766 (corresponding array for free polyominoids) for details.

			Array begins:
  n\k| 1  2  3   4   5    6     7      8      9      10       11        12
  ---+--------------------------------------------------------------------
   1 | 1  0  0   0   0    0     0      0      0       0        0         0
   2 | 1  1  1   1   1    1     1      1      1       1        1         1
   3 | 2  0  0   0   0    0     0      0      0       0        0         0
   4 | 2  2  2   2   2    2     2      2      2       2        2         2
   5 | 2  4 12  38 126  432  1520   5450  19820   72892   270536   1011722
   6 | 2  6 22  88 372 1628  7312  33466 155446  730534  3466170  16576874
   7 | 1  0  0   0   0    0     0      0      0       0        0         0
   8 | 1  2  6  19  63  216   760   2725   9910   36446   135268    505861
   9 | 1  2  6  19  63  216   760   2725   9910   36446   135268    505861
  10 | 1  4 20 110 638 3832 23592 147941 940982 6053180 39299408 257105146
  11 | 3  0  0   0   0    0     0      0      0       0        0         0
  12 | 3  3  3   3   3    3     3      3      3       3        3         3

Pontus von Brömssen, Python programs that relate row numbers and parameter sets.
Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A366766 (free), A366768.
The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C (see A366766). Some sequences occur in more than one row. Notation used in the table:
  X: Allowed connection.
  -: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).
  .: Not applicable for (D,d) in this row.
  !: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.
  *: Whether a connection of type (g,h) is allowed or not is independent of h.
      |   |   | connections |
      |   |   |  g:112223   |
    n | D | d |  h:010120   | sequence
  ----+---+---+-------------+----------
| 1 | 1 |  * -.....   | A063524
| 1 | 1 |  * X.....   | A000012
|!2 | 1 |  * --....   | 2*A063524
|!2 | 1 |    X-....   | 2*A000012
| 2 | 1 |    -X....   | 2*A001168
| 2 | 1 |  * XX....   | A096267
| 2 | 2 |  * -.-...   | A063524
| 2 | 2 |  * X.-...   | A001168
| 2 | 2 |  * -.X...   | A001168
| 2 | 2 |  * X.X...   | A006770
|!3 | 1 |  * --....   | 3*A063524
|!3 | 1 |    X-....   | 3*A000012
| 3 | 1 |    -X....   | A365655
| 3 | 1 |  * XX....   | A365560
|!3 | 2 |  * ----..   | 3*A063524
|!3 | 2 |    X---..   | 3*A001168
| 3 | 2 |    -X--..   | A365655
| 3 | 2 |  * XX--..   | A075678
|!3 | 2 |    --X-..   | 3*A001168
|!3 | 2 |    X-X-..   | 3*A006770
| 3 | 2 |    -XX-..   | A365996
| 3 | 2 |    XXX-..   | A365998
| 3 | 2 |    ---X..   | A366000
| 3 | 2 |    X--X..   | A366002
| 3 | 2 |    -X-X..   | A366004
| 3 | 2 |    XX-X..   | A366006
| 3 | 2 |  * --XX..   | A365653
| 3 | 2 |    X-XX..   | A366008
| 3 | 2 |    -XXX..   | A366010
| 3 | 2 |  * XXXX..   | A365651
| 3 | 3 |  * -.-..-   | A063524
| 3 | 3 |  * X.-..-   | A001931
| 3 | 3 |  * -.X..-   | A039742
| 3 | 3 |  * X.X..-   |
| 3 | 3 |  * -.-..X   | A039741
| 3 | 3 |  * X.-..X   |
| 3 | 3 |  * -.X..X   |
| 3 | 3 |  * X.X..X   |
|!4 | 1 |  * --....   | 4*A063524
|!4 | 1 |    X-....   | 4*A000012
| 4 | 1 |    -X....   | A366341
| 4 | 1 |  * XX....   | A365562
|!4 | 2 |  * -----.   | 6*A063524
|!4 | 2 |    X----.   | 6*A001168
| 4 | 2 |    -X---.   | A366339
| 4 | 2 |  * XX---.   | A366335
|!4 | 2 |    --X--.   | 6*A001168
|!4 | 2 |    X-X--.   | 6*A006770

A151830 Number of fixed 4-dimensional polycubes with n cells.

Original entry on oeis.org

1, 4, 28, 234, 2162, 21272, 218740, 2323730, 25314097, 281345096, 3178474308, 36400646766, 421693622520, 4933625049464, 58216226287844, 692095652493483

Offset: 1

N. J. A. Sloane, Jul 12 2009

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 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
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), 546-565.

G. Aleksandrowicz and G. Barequet, counting d-dimensional polycubes and nonrectangular planar polyomnoes, Lect. Not. Comp. Sci 4112 (2006) 418-427 Table 2
Gill Barequet, Gil Ben-Shachar, 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).

Cf. A001931, A068870 (free), A151831, A151832, A151833, A151834, A151835.

a(16) from Luther and Mertens by Gill Barequet, Jun 12 2011

A048663 Number of rooted polycubes with n cells, with no symmetries removed.

Original entry on oeis.org

1, 6, 45, 344, 2670, 20886, 164514, 1303304, 10375830, 82947380, 665440039, 5354470860, 43196001173, 349254823554, 2829388506690, 22961191276080, 186622811691276, 1518914831183982, 12377727000122043

Offset: 1

Richard C. Schroeppel, Dan Hoey

nonn

"Rooted" means some cell of the polycube is designated as the origin. This has the effect of multiplying the count by the volume of the polycube.

			There are six dicubes, each consisting of the origin cube together with one adjacent cube, in each of the six directions.

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, 2004.

A row of the array in A048790.

Cf. A001931.

A001931 = Cases[Import["https://oeis.org/A001931/b001931.txt", "Table"], {, }][[All, 2]];
a[n_] := n * A001931[[n]];
Array[a, 19] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = n * A001931(n). - Andrew Howroyd, Dec 04 2018

a(12)-a(19) from Andrew Howroyd, Dec 04 2018

A118356 Number of clusters with n vertices, n-1 edges and zero contacts on the simple cubic lattice.

Original entry on oeis.org

1, 3, 15, 83, 486, 2967, 18748, 121725, 807381, 5447203, 37264974, 257896500, 1802312605, 12701190885, 90157130289, 644022007040, 4626159163233

Offset: 1

R. J. Mathar, May 14 2006

a(n)<=A001931(n) due to the "no-contact" restriction.

An alternative wording for a(n) is the number of n-cell fixed tree-like polycubes in 3 dimensions. - Gill Barequet, May 25 2011

G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.

Gill Barequet, Gil Ben-Shachar, 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).
N. Madras, C. E. Soteros, S. G. Whittington, J. L. Martin et al., The free energy of a collapsing branched polymer, J Phys A: Math Gen 23 (1990) 5327-5350

Cf. A066158 (fixed tree-like polyominoes), A191094, A191095, A191096, A191097, A191098 (fixed tree-like polycubes in 4, 5, 6, 7, and 8 dimensions, resp.).

a(1)=1 added by Gill Barequet, May 25 2011

A151832 Number of fixed 6-dimensional polycubes with n cells.

Original entry on oeis.org

1, 6, 66, 901, 13881, 231008, 4057660, 74174927, 1398295989, 27012396022, 532327974882, 10665521789203, 216696065279573, 4455636282185802, 92567760074841818

Offset: 1

N. J. A. Sloane, Jul 12 2009

Anthony J. Guttmann, editor. Polygons, Polyominoes and Polycubes, volume 775 of Lecture Notes in Physics. Springer-Verlag, Heidelberg, 2009.

J. Adler, Y. Meir, A. B. Harris, A. Aharony, and J. A. M. S. Duarté, Series study of random animals in general dimensions, Physical Review B, 38 (1988) 4941.
Gadi Aleksandrowicz and Gill Barequet, Counting polycubes without the dimensionality curse, Discrete Mathematics, 309 (2009), 4576-4583.
Gadi Aleksandrowicz and Gill Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
Gadi Aleksandrowicz and Gill Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.
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. (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.
Ronnie Barequet, Gill Barequet, and Günter Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30, 257-275 (2010).
D. S. Gaunt and P. J. Peard, 1/d-expansions for the free energy of weakly embedded site animal models of branched polymers, Journal of Physics A: Mathematical and General, 33 (2000) 7515-7539.
Hsiao-Ping Hsu, Walter Nadler, and Peter Grassberger, Statistics of lattice animals, Computer Physics Communications, 169 (2005) 114-116.
Iwan Jensen, Enumerations of lattice animals and trees, Journal of Statistical Physics, 102(3/4) (2001) 865-881.
Sebastian Luther and Stephan Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565. See Table 2 for the terms (but beware of the incorrect a(13)!) or Table 3 for the formulas.
Sebastian Luther and Stephan Mertens, Counting lattice animals in high dimensions, arXiv:1106.1078 [cond-mat.stat-mech], 2011.
Stephan Mertens and Markus E. Lautenbacher, Counting lattice animals: A parallel attack, J. Stat. Phys. 66 (1992) 669.

Cf. A001931, A048667, A151830, A151831, A151833, A151834, A151835.

A048667 = Cases[Import["https://oeis.org/A048667/b048667.txt", "Table"], {, }][[All, 2]];
a[n_] := A048667[[n]]/n;
Array[a, 15] (* Jean-François Alcover, Sep 12 2019 *)

a(n) = A048667(n)/n. - Jean-François Alcover, Sep 12 2019, after Andrew Howroyd in A048667.

a(10) from Gadi Aleksandrowicz (gadial(AT)gmail.com), Mar 21 2010
a(11)-a(15) from Luther and Mertens by Gill Barequet, Jun 12 2011
a(13) corrected by M. F. Hasler, Jun 26 2025

A151833 Number of fixed 7-dimensional polycubes with n cells.

Original entry on oeis.org

1, 7, 91, 1484, 27468, 551313, 11710328, 259379101, 5933702467, 139272913892, 3338026689018, 81406063278113, 2014611366114053, 50486299825273271

Offset: 1

N. J. A. Sloane, Jul 12 2009

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
G. Aleksandrowicz and G. Barequet, Counting polycubes without the dimensionality curse, Discrete Mathematics, 309 (2009), 4576-4583.
G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 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
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), 546-565.

Gill Barequet, Gil Ben-Shachar, 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).

Cf. A001931, A048668, A151830, A151831, A151832, A151834, A151835.

A048668 = Cases[Import["https://oeis.org/A048668/b048668.txt", "Table"], {, }][[All, 2]];
a[n_] := A048668[[n]]/n;
Array[a, 14] (* Jean-François Alcover, Sep 12 2019 *)

a(n) = A048668(n)/n. - Jean-François Alcover, Sep 12 2019, after Andrew Howroyd in A048668.

More terms from Gadi Aleksandrowicz (gadial(AT)gmail.com), Mar 21 2010

a(11)-a(14) from Luther and Mertens by Gill Barequet, Jun 12 2011

A151834 Number of fixed 8-dimensional polycubes with n cells.

Original entry on oeis.org

1, 8, 120, 2276, 49204, 1156688, 28831384, 750455268, 20196669078, 558157620384, 15762232227968, 453181069339660, 13228272325440164, 391166062869849024

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.
G. Aleksandrowicz and G. Barequet, Counting polycubes without the dimensionality curse, Discrete Mathematics, 309 (2009), 4576-4583.
G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.
Gill Barequet, Gil Ben-Shachar, 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, A151835.

More terms from Gadi Aleksandrowicz (gadial(AT)gmail.com), Mar 21 2010
a(9)-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

A151831 Number of fixed 5-dimensional polycubes with n cells.

Original entry on oeis.org

1, 5, 45, 495, 6095, 80617, 1121075, 16177405, 240196280, 3648115531, 56440473990, 886696345225, 14111836458890, 227093585071305, 3689707621144614

Offset: 1

N. J. A. Sloane, Jul 12 2009

G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 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
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), 546-565.

G. Aleksandrowicz and G. Barequet, counting d-dimensional polycubes and nonrectangular planar polyomnoes, Lect. Not. Comp. Sci 4112 (2006) 418-427 Table 2
Gill Barequet, Gil Ben-Shachar, 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).

Cf. A001931, A048666, A151830, A151832, A151833, A151834, A151835.

A048666 = Cases[Import["https://oeis.org/A048666/b048666.txt", "Table"], {, }][[All, 2]];
a[n_] := A048666[[n]]/n;
Array[a, 15] (* Jean-François Alcover, Sep 12 2019 *)

a(n) = A048666(n)/n. - Jean-François Alcover, Sep 12 2019, after Andrew Howroyd in A048666.

a(14) and a(15) from Luther and Mertens by Gill Barequet, Jun 12 2011

cp's OEIS Frontend

A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

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A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells.

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A366767 Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

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A151830 Number of fixed 4-dimensional polycubes with n cells.

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A048663 Number of rooted polycubes with n cells, with no symmetries removed.

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A118356 Number of clusters with n vertices, n-1 edges and zero contacts on the simple cubic lattice.

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A151832 Number of fixed 6-dimensional polycubes with n cells.

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A151833 Number of fixed 7-dimensional polycubes with n cells.

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