A049430 -id:A049430 - cp's OEIS frontend

A005519 Let T(n,d) = number of distinct d-dimensional polyominoes (or polycubes) with n cells (A049429, A049430); sequence gives Sum_{d} T(n,d).

1, 1, 2, 7, 26, 153, 1134, 11050, 128987, 1765252, 27418060

Offset: 1

These are free polyominoes, meaning that they are counted up to rotation, reflection, and translation. - Peter Kagey, Apr 30 2020

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.
W. F. Lunnon, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Schroeppel, A few mathematical experiments

Row sums of A049429 or A049430.

a(9) from Richard C. Schroeppel

a(10) and a(11) from Peter Kagey, Apr 30 2020, based on the rows John Niss Hansen added to A049430

A330891 Triangle read by rows: cumulative sums of the rows of A049430.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 7, 0, 1, 12, 23, 26, 0, 1, 35, 112, 147, 153, 0, 1, 108, 607, 1019, 1123, 1134, 0, 1, 369, 3811, 8699, 10708, 11027, 11050, 0, 1, 1285, 25413, 82535, 119120, 127989, 128940, 128987, 0, 1, 4655, 178083, 846042, 1493722, 1725296

Offset: 1

Peter Kagey, Apr 30 2020

T(n,k) is also the number of n-celled polyominoes made up of k-dimensional cubes, counted up to rotation, reflection, and translation.

			Table begins:
n/k| 0 1    2     3     4      5      6      7      8
---+-------------------------------------------------
  1| 1
  2| 0 1
  3| 0 1    2
  4| 0 1    5     7
  5| 0 1   12    23    26
  6| 0 1   35   112   147    153
  7| 0 1  108   607  1019   1123   1134
  8| 0 1  369  3811  8699  10708  11027  11050
  9| 0 1 1285 25413 82535 119120 127989 128940 128987

Code Golf Stack Exchange, Counting hypercube Tetris pieces

Cf. A049429, A049430.
Columns 2-4: A000105, A038119, A068870.
Main diagonal is A005519.

T(n,k) = Sum_{i=0..k} A049430(n,i).

A068870 Number of polyhypercubes or 4-dimensional polyominoes with n cells (regarding mirror-images as identical).

Original entry on oeis.org

1, 1, 1, 2, 7, 26, 147, 1019, 8699, 82535, 846042, 9078720

Offset: 0

Franklin T. Adams-Watters, Mar 29 2002

Don Reble, C++ Program

Excluding a(0), 140th row of A366766.

Cf. A049429, A049430, A255487.

a(10) and a(11) from Don Reble, Feb 25 2015. - N. J. A. Sloane, Mar 01 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.

A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (n >= 1, 0 <= d <= n-1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 6, 3, 0, 1, 17, 17, 4, 0, 1, 59, 131, 52, 7, 0, 1, 195, 915, 709, 153, 13, 0, 1, 703, 6553, 8946, 3350, 454, 28

Offset: 1

N. J. A. Sloane, Sep 22 2011

From Petros Hadjicostas, Jan 11 2019: (Start)
Table 1 (p. 366) in Lunnon (1975) contains more terms. Because the table there (in the reference) has incomplete columns, the extra terms do not appear in this triangular sequence (array).
Entry DR(n=11, d=2) in Table 1 (p. 366) must be a typo. It should not be 33890, but 33895. This was corrected by N. J. A. Sloane in 2011 in the documentation of sequence A006758. (See also sequence A000988.)
(End)
The number of oriented polyominoes (chiral pairs counted as two) here is the sum of the number of unoriented polyominoes (chiral pairs counted as one) in A049430 and the number of chiral pairs. - Robert A. Russell, May 03 2020

			Triangle begins:
n\d| 0    1    2    3    4    5    6    7
---+---------------------------------=---
1  | 1
2  | 0    1
3  | 0    1    1
4  | 0    1    6    3
5  | 0    1   17   17    4
6  | 0    1   59  131   52    7
7  | 0    1  195  915  709  153   13
8  | 0    1  703 6553 8946 3350  454   28
...

W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (4) (1975) 366-367.

Columns give A006758 (and A000988), A006759, A006760, A006761.
Cf. A195739, A049430.
Cf. A000055, A045649, A036364, A036365.

From Robert A. Russell, May 03 2020: (Start)
For n > 1, DR(n,n-1) = A000055(n) + A045649(n).
DR(n,n-2) = A036364(n) + A036365(n).
We can add unoriented and chiral pairs for the top two diagonals. The summands have quick algorithms. (End)

Sequence corrected by Petros Hadjicostas, Jan 11 2019 after observation by Jon E. Schoenfield

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.

A355053 Number of unoriented multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 11, 77, 412, 2009, 8869, 36988, 146578, 560498, 2078927, 7530385, 26734692, 93360884, 321454484, 1093599885, 3681897625, 12284317088, 40660245162, 133638662066, 436488290069, 1417680926923, 4581355626106

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). For unoriented polyominoes, chiral pairs are counted as one.

			a(4)=1 as there is only one tetromino in one-space. a(5)=11 because there are 5 achiral and 6 chiral pairs of pentominoes in 2-space, excluding the 1-D straight pentomino.

Robert A. Russell, Table of n, a(n) for n = 4..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), A355054 (chiral), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A000081 (rooted trees), A049430 (Lunnon's DE).

Other dimensions: A036364 (n-2), A000055 (n-1), A355048 (orthoplex).

sb[n_,k_]:= sb[n,k] = b[n+1-k,1] + If[n<2k, 0, sb[n-k,k]];
b[1,1] := 1; b[n_,1] := b[n,1] = Sum[b[i,1]sb[n-1,i]i,{i,1,n-1}]/(n-1);
b[n_,k_] := b[n,k] = Sum[b[i,1]b[n-i,k-1],{i,1,n-1}];
nmax = 30; B[x_] := Sum[b[i,1]x^i,{i,0,nmax}]
Drop[CoefficientList[Series[(50B[x]^6+3B[x]^7+30B[x]^2B[x^2]^2+3B[x]^3B[x^2](6+B[x^2])+3B[x]^5(37+2B[x^2])+12B[x]^4(1+3B[x^2])+B[x](57B[x^2]^2+6B[x^2]^3+6B[x^4]+6B[x^2]B[x^4])+4(3B[x^2]^2+11B[x^2]^3+B[x^3]^2+B[x^6]))/24+B[x]^2(112B[x]^5+9B[x]^6+3B[x^2]^2+4B[x]B[x^2]^2+B[x]^2B[x^2](14+B[x^2])+8B[x]^3(1+4B[x^2])+B[x]^4(167+10B[x^2]))/(8(1-B[x]))+B[x]^5(46B[x]^3+6B[x]^4+3B[x^2]+B[x]^2(67+2B[x^2])+B[x](2+6B[x^2]))/(2(1-B[x])^2)+B[x]^6(153B[x]^2+75B[x]^3+12B[x]^4+3B[x^2]+B[x](4+3B[x^2]))/(6(1-B[x])^3)+B[x]^9(21+4B[x])/(2(1-B[x])^4)+3B[x]^10/(2(1-B[x])^5)+B[x^2](6B[x]^3B[x^2]+2B[x]^4B[x^2]+13B[x^2]^2+19B[x^2]^3+2B[x]^2B[x^2](1+3B[x^2])+B[x^4]+B[x^2]B[x^4]+B[x](35B[x^2]^2+5B[x^2]^3+B[x^4]+B[x^2]B[x^4]))/(4(1-B[x^2]))+B[x^2]^4(5+3B[x^2]+B[x](8+B[x^2]))/(1-B[x^2])^2+2B[x^2]^5(1+B[x])/(1-B[x^2])^3+2B[x]B[x^3]^2/(6(1-B[x^3]))+B[x]B[x^4]^2/(2(1-B[x^4]))+B[x]^2B[x^2]^2(7B[x]^2+5B[x]^3+3B[x^2]+B[x](2+B[x^2]))/(2(1-B[x])(1-B[x^2]))+B[x]^5B[x^2]^2(3+2B[x])/((1-B[x])^2(1-B[x^2]))+B[x]^6B[x^2]^2/((1-B[x])^3(1-B[x^2]))+B[x]^2B[x^2]^4/((1-B[x])(1-B[x^2])^2)+B[x^2]B[x^4]^2(1+B[x])/(2(1-B[x^2])(1-B[x^4])),{x,0,nmax}],x],4]

a(n) = A355052(n) - A355054(n) = (A355052(n) + A355055(n)) / 2 = A355054(n) + A355055(n).
a(n) = A049430(n,n-3), the third diagonal of Lunnon's DE array.
G.f.: (50B(x)^6+3B(x)^7+30B(x)^2B(x^2)^2+3B(x)^3B(x^2)(6+B(x^2))+3B(x)^5(37+2B(x^2))+12B(x)^4(1+3B(x^2))+B(x)(57B(x^2)^2+6B(x^2)^3+6B(x^4)+6B(x^2)B(x^4))+4(3B(x^2)^2+11B(x^2)^3+B(x^3)^2+B(x^6)))/24 + B(x)^2(112B(x)^5+9B(x)^6+3B(x^2)^2+4B(x)B(x^2)^2+B(x)^2B(x^2)(14+B(x^2))+8B(x)^3(1+4B(x^2))+B(x)^4(167+10B(x^2)))/(8(1-B(x))) + B(x)^5(46B(x)^3+6B(x)^4+3B(x^2)+B(x)^2(67+2B(x^2))+B(x)(2+6B(x^2)))/(2(1-B(x))^2) + B(x)^6(153B(x)^2+75B(x)^3+12B(x)^4+3B(x^2)+B(x)(4+3B(x^2)))/(6(1-B(x))^3) + B(x)^9(21+4B(x))/(2(1-B(x))^4) + 3B(x)^10/(2(1-B(x))^5) + B(x^2)(6B(x)^3B(x^2)+2B(x)^4B(x^2)+13B(x^2)^2+19B(x^2)^3+2B(x)^2B(x^2)(1+3B(x^2))+B(x^4)+B(x^2)B(x^4)+B(x)(35B(x^2)^2+5B(x^2)^3+B(x^4)+B(x^2)B(x^4)))/(4(1-B(x^2))) + B(x^2)^4(5+3B(x^2)+B(x)(8+B(x^2)))/(1-B(x^2))^2 + 2B(x^2)^5(1+B(x))/(1-B(x^2))^3 + 2B(x)B(x^3)^2/(6(1-B(x^3))) + B(x)B(x^4)^2/(2(1-B(x^4))) + B(x)^2B(x^2)^2(7B(x)^2+5B(x)^3+3B(x^2)+B(x)(2+B(x^2)))/(2(1-B(x))(1-B(x^2))) + B(x)^5B(x^2)^2(3+2B(x))/((1-B(x))^2(1-B(x^2))) + B(x)^6B(x^2)^2/((1-B(x))^3(1-B(x^2))) + B(x)^2B(x^2)^4/((1-B(x))(1-B(x^2))^2) + B(x^2)B(x^4)^2(1+B(x))/(2(1-B(x^2))(1-B(x^4))), where B(x) is the generating function for rooted trees with n nodes in A000081.

A365566 Triangle read by rows: T(n,d) is the number of inequivalent properly d-dimensional n-polysticks (or polyedges), 1 <= d <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 15, 12, 3, 1, 54, 105, 39, 6, 1, 221, 863, 566, 117, 11

Offset: 1

Pontus von Brömssen, Sep 09 2023

			Triangle begins:
  n\d | 1   2   3   4   5  6
  ----+---------------------
   1  | 1
   2  | 1   1
   3  | 1   4   2
   4  | 1  15  12   3
   5  | 1  54 105  39   6
   6  | 1 221 863 566 117 11

Index entries for sequences related to polyominoes.

Cf. A000055, A049430 (polyominoes), A365565 (row sums), A385582 (fixed), A385583.

T(n,n) = A000055(n+1).

T(n,d) = A385583(n,d) - A385583(n,d-1) (with A385583(n,0) = 0). - Pontus von Brömssen, Jul 13 2025

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.

A006765 Number of strictly 2-dimensional polyominoes with n cells.

Original entry on oeis.org

0, 0, 1, 4, 11, 34, 107, 368, 1284, 4654, 17072, 63599, 238590, 901970, 3426575, 13079254, 50107908, 192622051, 742624231, 2870671949, 11123060677, 43191857687, 168047007727, 654999700402, 2557227044763, 9999088822074, 39153010938486, 153511100594602, 602621953061977, 2368347037571251, 9317706529987949, 36695016991712878

Offset: 1

N. J. A. Sloane

nonn

A000105 is the main entry for this sequence.

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.
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 = 1..50 (terms 1..45 from N. J. A. Sloane)
W. F. Lunnon, Counting multidimensional polyominoes. Computer Journal 18 (1975), no. 4, pp. 366-367.
R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, California, March 30, 2004.

Equals A000105(n)-1. A row of A049429. A column of A049430.

Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]] - 1 // Rest (* Jean-François Alcover, Jan 21 2020 *)

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

Name clarified by John Mason, Jan 10 2023

cp's OEIS Frontend

A005519 Let T(n,d) = number of distinct d-dimensional polyominoes (or polycubes) with n cells (A049429, A049430); sequence gives Sum_{d} T(n,d).

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A330891 Triangle read by rows: cumulative sums of the rows of A049430.

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A068870 Number of polyhypercubes or 4-dimensional polyominoes with n cells (regarding mirror-images as identical).

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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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A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (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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A355053 Number of unoriented multidimensional n-ominoes with cell centers determining n-3 space.

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A365566 Triangle read by rows: T(n,d) is the number of inequivalent properly d-dimensional n-polysticks (or polyedges), 1 <= 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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