A195738 -id:A195738 - cp's OEIS frontend

A000988 Number of one-sided polyominoes with n cells.

1, 1, 1, 2, 7, 18, 60, 196, 704, 2500, 9189, 33896, 126759, 476270, 1802312, 6849777, 26152418, 100203194, 385221143, 1485200848, 5741256764, 22245940545, 86383382827, 336093325058, 1309998125640, 5114451441106, 19998172734786, 78306011677182, 307022182222506, 1205243866707468, 4736694001644862

Offset: 0

N. J. A. Sloane, hugh(AT)mimosa.com (D. Hugh Redelmeier)

nonn

A000105(n) + A030228(n) = a(n) because the number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of one-sided polyominoes. - Graeme McRae, Jan 05 2006

Names for the first few polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, enneomino (aka nonomino), decomino, hendecomino (aka undecomino), dodecomino, ...

			a(0) = 1 as there is 1 empty polyomino with #cells = 0. - _Fred Lunnon_, Jun 24 2020

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
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).

John Mason, Table of n, a(n) for n = 0..50 (terms 0..45,47,49 from Toshihiro Shirakawa).
W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18(4) (1975), 366-367.
Ed Pegg, Jr., Illustrations of polyforms.
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9(3) (2005), 609-640. [Broken link]
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9(3) (2005), 609-640. [From the internet archive]
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Harmonic Magic Square, pp. 3-4: Enumeration of Polyominoes considering the symmetry, April 2012.
Eric Weisstein's World of Mathematics, Polyomino.
Wikipedia, Polyomino.

See A006758 for another version. Subtracting 1 gives first column of A195738. Cf. A000105 (unoriented), A030228 (chiral), A030227 (achiral), A001168 (fixed).

a(n) = 2*A006749(n) + A006746(n) + A006748(n) + 2*A006747(n) + A056877(n) + A056878(n) + 2*A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

a(n) = 2*A000105(n) - A030227(n) = 2*A030228(n) + A030227(n). - Robert A. Russell, Feb 03 2022

a(0) = 1 added by N. J. A. Sloane, Jun 24 2020

A049430 Triangle read by rows: T(n,d) is the number of distinct properly d-dimensional polyominoes (or polycubes) with n cells (n >= 1, d >= 0).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 2, 0, 1, 11, 11, 3, 0, 1, 34, 77, 35, 6, 0, 1, 107, 499, 412, 104, 11, 0, 1, 368, 3442, 4888, 2009, 319, 23, 0, 1, 1284, 24128, 57122, 36585, 8869, 951, 47, 0, 1, 4654, 173428, 667959, 647680, 231574, 36988, 2862, 106, 0, 1, 17072, 1262464, 7799183, 11173880, 5712765, 1297366, 146578, 8516, 235

Offset: 1

Richard C. Schroeppel

			Triangle begins:
1
0 1
0 1     1
0 1     4       2
0 1    11      11       3
0 1    34      77      35        6
0 1   107     499     412      104      11
0 1   368    3442    4888     2009     319      23
0 1  1284   24128   57122    36585    8869     951     47
0 1  4654  173428  667959   647680  231574   36988   2862  106
0 1 17072 1262464 7799183 11173880 5712765 1297366 146578 8516 235
...

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

Cf. A049429 (col. d=0 omitted), A195738 (oriented), A195739 (fixed).
Row sums give A005519. Columns give A006765, A006766, A006767, A006768.
Diagonals (with algorithms) are A000055, A036364, A355053.
Cf. A330891 (cumulative sums of the rows).

Edited by N. J. A. Sloane, Sep 23 2011

More terms from John Niss Hansen, Mar 31 2015

A195739 Triangle read by rows: DX(n,d) = number of properly d-dimensional polyominoes with n cells, modulo translations (n>=1, 0 <= d <= n-1).

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 0, 1, 17, 32, 0, 1, 61, 348, 400, 0, 1, 214, 2836, 8640, 6912, 0, 1, 758, 21225, 129288, 254800, 153664, 0, 1, 2723, 154741, 1688424, 6160640, 8749056, 4194304, 0, 1, 9908, 1123143, 20762073, 125055400, 313921008, 343901376, 136048896

Offset: 1

N. J. A. Sloane, Sep 23 2011

According to Barequet-Barequet-Rote, p. 261, the value DX(7, 6) = 134209 given by W. F. Lunnon is incorrect; it should be 153664, see A127670. - Alexander Knapp, May 13 2013

			Triangle begins with DX(1,0):
n\d 0  1   2     3      4      5      6
---------------------------------------
1...1
2...0  1
3...0  1   4
4...0  1  17    32
5...0  1  61   348    400
6...0  1 214  2836   8640   6912
7...0  1 758 21225 129288 254800 153664
...

Robert A. Russell, Table of n, a(n) for n = 1..60
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (1975), no. 4, pp. 366-367.

Columns give A006762, A006763, A006764. Cf. A195738, A049430.

Diagonals (with formulas) are A127670, A171860, A191092, A259015, A290738.

A049429 Triangle T(n,d) = number of distinct d-dimensional polyominoes (or polycubes) with n cells (0 < d < n).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 11, 11, 3, 1, 34, 77, 35, 6, 1, 107, 499, 412, 104, 11, 1, 368, 3442, 4888, 2009, 319, 23, 1, 1284, 24128, 57122, 36585, 8869, 951, 47, 1, 4654, 173428, 667959, 647680, 231574, 36988, 2862, 106, 1, 17072, 1262464, 7799183, 11173880, 5712765, 1297366, 146578, 8516, 235

Offset: 2

Richard C. Schroeppel

These are unoriented polyominoes of the regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). For unoriented polyominoes, chiral pairs are counted as one. The dimension of the convex hull of the cell centers determines the dimension d. - Robert A. Russell, Aug 09 2022

			From _Robert A. Russell_, Aug 09 2022: (Start)
Triangle begins with T(2,1):
n\d 1     2       3       4        5       6       7      8    9  10
2   1
3   1     1
4   1     4       2
5   1    11      11       3
6   1    34      77      35        6
7   1   107     499     412      104      11
8   1   368    3442    4888     2009     319      23
9   1  1284   24128   57122    36585    8869     951     47
10  1  4654  173428  667959   647680  231574   36988   2862  106
11  1 17072 1262464 7799183 11173880 5712765 1297366 146578 8516 235
(End)

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

Cf. A049430 (col. d=0 added), A195738 (oriented), A195739 (fixed).
Diagonals (with algorithms) are A000055, A036364, A355053.
Row sums give A005519. Columns are A006765-A006768.

Two more rows added by Robert A. Russell, Aug 09 2022.

A355054 Number of chiral pairs of multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

6, 54, 297, 1341, 5468, 20519, 72168, 242886, 791780, 2514453, 7814225, 23863941, 71835845, 213601046, 628450974, 1832227629, 5299559865, 15221688836, 43450246045, 123345029035, 348417524877, 979803281560

Offset: 5

Robert A. Russell, Jun 16 2022

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). Each member of a chiral pair is a reflection but not a rotation of the other.

			a(5)=6 because there are 6 chiral pairs of pentominoes in 2-space.

Robert A. Russell, Table of n, a(n) for n = 5..100
W. F. Lunnon, Counting multidimensional polyominoes. Computer Journal 18 (1975), no. 4, pp. 366-367.
Robert A. Russell, Trunk Generating Functions

Cf. A355052 (oriented), A355053 (unoriented), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A045648 (rooted chiral), A195738 (Lunnon's DR), A049430 (Lunnon's DE).

Other dimensions: A036365 (n-2), A045649 (n-1), A355049 (orthoplex).

sc[n_,k_] := sc[n,k] = c[n+1-k,1] + If[n<2k, 0, sc[n-k,k](-1)^k];
c[1,1] := 1; c[n_,1] := c[n,1] = Sum[c[i,1] sc[n-1,i]i, {i,1,n-1}]/(n-1);
c[n_,k_] := c[n, k] = Sum[c[i, 1] c[n-i, k-1], {i,1,n-1}];
nmax = 30; K[x_] := Sum[c[i,1] x^i, {i,0,nmax}]
Drop[CoefficientList[Series[(12 K[x]^4 + 87 K[x]^5 + 50 K[x]^6 + 3 K[x]^7 + 18 K[x]^3 K[-x^2] + 36 K[x]^4 K[-x^2] + 6 K[x]^5 K[-x^2] - 12 K[-x^2]^2 - 27 K[x] K[-x^2]^2 - 6 K[x]^2 K[-x^2]^2 + 3 K[x]^3 K[-x^2]^2 - 16 K[-x^2]^3 - 6 K[x] K[-x^2]^3 + 4 K[x^3]^2 - 6 K[x] K[-x^4] - 6 K[x] K[-x^2] K[-x^4] + 4 K[-x^6]) / 24 + K[x]^2 (16 K[x]^3 + 159 K[x]^4 + 112 K[x]^5 + 9 K[x]^6 + 14 K[x]^2 K[-x^2] + 32 K[x]^3 K[-x^2] + 10 K[x]^4 K[-x^2] - K[-x^2]^2 + K[x]^2 K[-x^2]^2) / (8 (1-K[x])) + K[x]^5 (2 K[x] + 67 K[x]^2 + 46 K[x]^3 + 6 K[x]^4 + 3 K[-x^2] + 6 K[x] K[-x^2] + 2 K[x]^2 K[-x^2]) / (2 (1-K[x])^2) - K[-x^2] (2 K[x]^2 K[-x^2] + 7 K[-x^2]^2 + 17 K[x] K[-x^2]^2 + 2 K[x]^2 K[-x^2]^2 + 7 K[-x^2]^3 + 5 K[x] K[-x^2]^3 + K[-x^4] + K[x] K[-x^4] + K[-x^2] K[-x^4] + K[x] K[-x^2] K[-x^4]) / (4 (1-K[-x^2])) + K[x]^6 (4 K[x] + 153 K[x]^2 + 75 K[x]^3 + 12 K[x]^4 + 3 K[-x^2] + 3 K[x] K[-x^2]) / (6 (1-K[x])^3) - K[x]^2 K[-x^2]^2 (K[x] + K[-x^2]) / ((1-K[x]) (1-K[-x^2])) + (K[x] K[x^3]^2) / (3 (1-K[x^3])) + K[x]^9 (21 + 4 K[x]) / (2 (1-K[x])^4) - K[-x^2]^4 (6 + 7 K[x] + 2 K[-x^2] + 2 K[x] K[-x^2]) / (2 (1-K[-x^2])^2) + 3 K[x]^10 / (2 (1-K[x])^5) - K[x]^2 K[-x^2]^4 / (2 (1-K[x]) (1-K[-x^2])^2) - (1 + K[x]) K[-x^2]^5 / (1-K[-x^2])^3, {x,0,nmax}], x], 5]

a(n) = A355052(n) - A355053(n) = (A355052(n) - A355055(n)) / 2 = A355053(n) - A355055(n).
a(n) = A195738(n,n-3) - A049430(n,n-3), diagonals of Lunnon's DR and DE arrays.
G.f.: (12 C(x)^4 + 87 C(x)^5 + 50 C(x)^6 + 3 C(x)^7 + 18 C(x)^3 C(-x^2) + 36 C(x)^4 C(-x^2) + 6 C(x)^5 C(-x^2) - 12 C(-x^2)^2 - 27 C(x) C(-x^2)^2 - 6 C(x)^2 C(-x^2)^2 + 3 C(x)^3 C(-x^2)^2 - 16 C(-x^2)^3 - 6 C(x) C(-x^2)^3 + 4 C(x^3)^2 - 6 C(x) C(-x^4) - 6 C(x) C(-x^2) C(-x^4) + 4 C(-x^6)) / 24 + C(x)^2 (16 C(x)^3 + 159 C(x)^4 + 112 C(x)^5 + 9 C(x)^6 + 14 C(x)^2 C(-x^2) + 32 C(x)^3 C(-x^2) + 10 C(x)^4 C(-x^2) - C(-x^2)^2 + C(x)^2 C(-x^2)^2) / (8 (1-C(x))) + C(x)^5 (2 C(x) + 67 C(x)^2 + 46 C(x)^3 + 6 C(x)^4 + 3 C(-x^2) + 6 C(x) C(-x^2) + 2 C(x)^2 C(-x^2)) / (2 (1-C(x))^2) - C(-x^2) (2 C(x)^2 C(-x^2) + 7 C(-x^2)^2 + 17 C(x) C(-x^2)^2 + 2 C(x)^2 C(-x^2)^2 + 7 C(-x^2)^3 + 5 C(x) C(-x^2)^3 + C(-x^4) + C(x) C(-x^4) + C(-x^2) C(-x^4) + C(x) C(-x^2) C(-x^4)) / (4 (1-C(-x^2))) + C(x)^6 (4 C(x) + 153 C(x)^2 + 75 C(x)^3 + 12 C(x)^4 + 3 C(-x^2) + 3 C(x) C(-x^2)) / (6 (1-C(x))^3) - C(x)^2 C(-x^2)^2 (C(x) + C(-x^2)) / ((1-C(x)) (1-C(-x^2))) + (C(x) C(x^3)^2) / (3 (1-C(x^3))) + C(x)^9 (21 + 4 C(x)) / (2 (1-C(x))^4) - C(-x^2)^4 (6 + 7 C(x) + 2 C(-x^2) + 2 C(x) C(-x^2)) / (2 (1-C(-x^2))^2) + 3 C(x)^10 / (2 (1-C(x))^5) - C(x)^2 C(-x^2)^4 / (2 (1-C(x)) (1-C(-x^2))^2) - (1 + C(x)) C(-x^2)^5 / (1-C(-x^2))^3 where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

A125761 Triangle read by rows: T(n,k) (n>=1) gives the number of n-indecomposable polyominoes with k cells (k >= 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 12, 6, 5, 1, 1, 1, 1, 2, 5, 12, 35, 108, 73, 76, 80, 25, 15, 15, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 1044, 1475, 2205, 2643, 983, 1050, 1208, 958, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 15980, 26548, 48766, 79579, 99860, 45898, 60433, 89890, 109424, 84312, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 245955, 458397, 948201, 1857965, 3160371, 4153971, 2217787, 3402761, 5855953, 9067535, 11402651, 9170285, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 3807508, 7710844, 17354771, 37983463

Offset: 1

David Applegate and N. J. A. Sloane, Feb 05 2007

A polyomino is called n-indecomposable if it cannot be partitioned (along cell boundaries) into two or more polyominoes each with at least n cells.
Row n has 4n-3 nonzero terms.
For full lists of drawings of these polyominoes for n <= 6, see the links in A125759.
Rows converge to A000105. - Andrey Zabolotskiy, Dec 26 2017

			Triangle begins:
1;
1,1,2,1,1;
1,1,2,5,12,6,5,1,1;
1,1,2,5,12,35,108,73,76,80,25,15,15;
1,1,2,5,12,35,108,369,1285,1044,1475,2205,2643,983,1050,1208,958;
1,1,2,5,12,35,108,369,1285,4655,17073,15980,26548,48766,79579,99860,45898,60433,89890,109424,84312;
1,1,2,5,12,35,108,369,1285,4655,17073,63600,238591,245955,458397,948201,1857965,3160371,4153971,2217787,3402761,5855953,9067535,11402651,9170285;
1,1,2,5,12,35,108,369,1285,4655,17073,63600,238591,901971,3426576,3807508,7710844,17354771,37983463,...

N. MacKinnon, Some thoughts on polyomino tilings, Math. Gaz., 74 (1990), 31-33.
Simone Rinaldi and D. G. Rogers, Indecomposability: polyominoes and polyomino tilings, The Mathematical Gazette 92.524 (2008): 193-204.

Row sums give A125759.

Cf. A125709, A125753, A126742, A126743; A000105, A195738, A195739, A049430.

Rows 5, 6, 7 and 8 from David Applegate, Feb 16 2007

A355052 Number of oriented multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 17, 131, 709, 3350, 14337, 57507, 218746, 803384, 2870707, 10044838, 34548917, 117224825, 393290329, 1307200931, 4310348599, 14116544717, 45959805027, 148860350902, 479938536114, 1541025955958, 4929773150983

Offset: 4

Robert A. Russell, Jun 16 2022

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). This sequence is obtained using the first formula below. For oriented polyominoes, chiral pairs are counted as two.

			a(4)=1 because there is just one tetromino (with four cells aligned) in 1-space. a(5)=17 because there are 5 achiral and 6 chiral pairs of pentominoes in 2-space, excluding the 1-D straight pentomino.

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

Cf. A355053 (unoriented), A355054 (chiral), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A355047 (orthoplex), A195738 (Lunnon's DR).

a(n) = A355053(n) + A355054(n) = 2*A355053(n) - A355055(n) = 2*A355054(n) + A355055(n).

a(n) = A195738(n,n-3), the third diagonal of Lunnon's DR array.

A355055 Number of achiral multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 5, 23, 115, 668, 3401, 16469, 74410, 317612, 1287147, 5015932, 18920467, 69496943, 249618639, 879998839, 3053446651, 10452089459, 35360685297, 118416973230, 393038044024, 1294335897888, 4232938101229, 13757913332396

Offset: 4

Robert A. Russell, Jun 16 2022

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). This sequence is obtained using the first formula below. An achiral polyomino is identical to its reflection.

			a(4)=1 as there is only one tetromino in one-space. a(5)=5 because there are 5 achiral pentominoes in 2-space, excluding the 1-D straight pentomino.

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

Cf. A355052 (oriented), A355053 (unoriented), A355054 (chiral), A355056 (asymmetric), A191092 (fixed), A355050 (orthoplex), A195738 (Lunnon's DR), A049430 (Lunnon's DE).

a(n) = A355053(n) - A355054(n) = 2*A355053(n) - A355052(n) = A355052(n) - 2*A355054(n).

a(n) = 2*A049430(n,n-3) - A195738(n,n-3), Lunnon's DE and DR arrays.

A006758 Number of strictly 2-dimensional one-sided polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 1, 6, 17, 59, 195, 703, 2499, 9188, 33895, 126758, 476269, 1802311, 6849776, 26152417, 100203193, 385221142, 1485200847, 5741256763, 22245940544, 86383382826, 336093325057, 1309998125639, 5114451441105, 19998172734785, 78306011677181, 307022182222505, 1205243866707467, 4736694001644861, 18635412907198669

Offset: 0

N. J. A. Sloane

nonn

A000988 is the main entry for this sequence.
This version counts only polyominoes that are strictly 2-dimensional - it excludes those where all the squares are in a single line. Thus a(n) = A000988(n) - 1. - John Mason, Feb 18 2021
The value of a(11) in Table 1 (p. 366) of Lunnon (1975) is incorrect, and was corrected here by N. J. A. Sloane in 2011. - Petros Hadjicostas, Jan 11 2019

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

N. J. A. Sloane, Table of n, a(n) for n = 0..45 [a(0)=0 inserted by _Georg Fischer_, Jun 25 2020]
W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (4) (1975) 366-367.

A000988 (= a(n) + 1) is another version. A column of A195738.

Edited by N. J. A. Sloane, Sep 23 2011
a(0)=0 inserted by Sean A. Irvine, Jun 24 2020
Name clarified by John Mason, Feb 18 2021

A006759 Number of one-sided strictly 3-dimensional polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 3, 17, 131, 915, 6553, 47026, 341888, 2505449, 18534827, 138224058, 1038594326, 7856087894, 59782042225, 457359506070, 3515816578512, 27143401299351, 210372490707568

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 (4) (1975) 366-367.

A column of A195738.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000105 = A@000105;
A000162 = A@000162;
a[n_] := A000162[[n]] - A000105[[n + 1]];
a /@ Range[16] (* Jean-François Alcover, Jan 16 2020 *)

a(11) corrected and a(13)-a(16) from A000162-A000105 by Jean-François Alcover, Jan 16 2020

Name clarified and a(17)-a(20) from A000162-A000105 by John Mason, Dec 26 2023

cp's OEIS Frontend

A000988 Number of one-sided polyominoes with n cells.

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A049430 Triangle read by rows: T(n,d) is the number of distinct properly d-dimensional polyominoes (or polycubes) with n cells (n >= 1, d >= 0).

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A195739 Triangle read by rows: DX(n,d) = number of properly d-dimensional polyominoes with n cells, modulo translations (n>=1, 0 <= d <= n-1).

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A049429 Triangle T(n,d) = number of distinct d-dimensional polyominoes (or polycubes) with n cells (0 < d < n).

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A355054 Number of chiral pairs of multidimensional n-ominoes with cell centers determining n-3 space.

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A125761 Triangle read by rows: T(n,k) (n>=1) gives the number of n-indecomposable polyominoes with k cells (k >= 1).

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A355052 Number of oriented multidimensional n-ominoes with cell centers determining n-3 space.

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A355055 Number of achiral multidimensional n-ominoes with cell centers determining n-3 space.

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A006758 Number of strictly 2-dimensional one-sided polyominoes with n cells.

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