A049430 -id:A049430 - cp's OEIS frontend

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

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

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.

A387004 Triangle read by rows: T(n,d) is the number of free, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

Original entry on oeis.org

1, 1, 1, 2, 7, 3, 5, 49, 41, 8

Offset: 1

Pontus von Brömssen, Aug 14 2025

			Triangle begins:
  n\d| 2  3  4  5
  ---+-----------
  1  | 1
  2  | 1  1
  3  | 2  7  3
  4  | 5 49 41  8

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

Cf. A000105 (column d=2), A049430 (polyominoes), A365566 (polysticks), A387002 (fixed), A387003, A387005 (row sums).

T(n,d) = A387003(n,d) - A387003(n,d-1) (with A387003(n,1) = 0).

A006766 Number of 3-dimensional polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 2, 11, 77, 499, 3442, 24128, 173428, 1262464, 9307494

Offset: 1

N. J. A. Sloane

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).

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.

A row of A049429. A column of A049430.

A006768 Number of 5-dimensional polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 104, 2009, 36585, 647680, 11173880

Offset: 1

N. J. A. Sloane

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).

W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (1975), no. 4, pp. 366-367.
R. C. Schroeppel, A few mathematical experiments

A row of A049429. A column of A049430.

a(10)-a(11) from Sean A. Irvine, Jul 29 2017

A255487 Number of polyhypercubes or 4-dimensional polyominoes with n cells (regarding mirror-images as distinct).

Original entry on oeis.org

1, 1, 1, 2, 7, 27, 164, 1316, 12757, 134174, 1474341, 16588434

Offset: 0

N. J. A. Sloane, Mar 01 2015

Don Reble, Personal communication, Feb 25 2015

Cf. A049429, A049430, A068870.

a(n) = A006760(n) + A006765(n) + A006766(n) + signum(n-1) for n >= 1. - Sean A. Irvine, Jul 19 2017

A365141 List of free 5-dimensional polyominoes in binary code (see A365139), ordered first by the number of cells, then by the value of the binary code.

Original entry on oeis.org

1, 3, 7, 67, 15, 71, 135, 139, 195, 198, 2097219, 31, 79, 143, 155, 199, 203, 327, 454, 653, 707, 710, 775, 902, 5127, 2097223, 2097347, 2097350, 4194375, 4194379, 4194693, 4194694, 4194756, 4194950, 4194954, 12583046, 72057594040025155, 63, 95, 159, 187, 207

Offset: 1

Pontus von Brömssen, Aug 23 2023

Can be read as an irregular triangle, whose n-th row contains Sum_{d=0..5} A049430(n,d) terms.

			As an irregular triangle:
  1;
  3;
  7, 67;
  15, 71, 135, 139, 195, 198, 2097219;
  ...

Index entries for sequences related to polyominoes.

Cf. A049430, A246521 (2 dimensions), A365139 (3 dimensions), A365140 (4 dimensions).

A006767 Number of 4-dimensional polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 0, 3, 35, 412, 4888, 57122, 667959, 7799183

Offset: 1

N. J. A. Sloane

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).

W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.
R. C. Schroeppel, A few mathematical experiments

A row of A049429. A column of A049430.

A290305 Number of 5-dimensional polyominoes with n cells (regarding mirror-images as distinct).

Original entry on oeis.org

1, 1, 1, 2, 7, 26, 154, 1172, 12049, 148508, 2061893, 30545504

Offset: 0

Sean A. Irvine, Jul 26 2017

Cf. A049429, A049430, A006761 (one-sided), A255487 (4-dimensional case).

a(n) = A006761(n) + A068870(n).

A365143 Proper dimension of the polyomino with code A365142(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 2, 2, 3, 1, 3, 2, 3, 3, 4, 4, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 2, 3, 3, 4, 1, 2, 3, 4, 4, 4, 3, 2, 3, 3, 2, 2, 2, 3, 3, 3, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 5, 5, 3, 2, 3, 3, 4, 4, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3

Offset: 1

Pontus von Brömssen, Aug 25 2023

Can be read as an irregular triangle, whose n-th row contains A005519(n) terms. The first term of the n-th row is A000720(n). The number of times d occurs in the n-th row is A049430(n,d).

			As an irregular triangle:
  0;
  1;
  2, 1;
  2, 3, 2, 2, 2, 3, 1;
  3, 2, 3, 3, 4, 4, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 2, 3, 3, 4, 1, 2;
  ...
For the 4th row, the seven 4-cell polyominoes, with codes 15, 23, 39, 43, 46, 51, 139 (4th row of A365142), are the L-tetromino, the properly 3-dimensional nonchiral tetracube, the square tetromino, the T-tetromino, the S-tetromino, the properly 3-dimensional chiral tetracube, and the straight tetromino, with proper dimensions 2, 3, 2, 2, 2, 3, 1, respectively.

Index entries for sequences related to polyominoes.

Cf. A000720, A005519, A049430, A061395, A133457, A365142.

a(n) = max_{1<=i<=m} A061395(e_i+1), where A365142(n) = Sum_{1<=i<=m} 2^e_i and e_1 < ... < e_m != 0 (i.e., (e_1, ..., e_m) is the A365142(n)-th row of A133457).

cp's OEIS Frontend

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

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A387004 Triangle read by rows: T(n,d) is the number of free, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

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A006766 Number of 3-dimensional polyominoes with n cells.

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A006768 Number of 5-dimensional polyominoes with n cells.

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A255487 Number of polyhypercubes or 4-dimensional polyominoes with n cells (regarding mirror-images as distinct).

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A365141 List of free 5-dimensional polyominoes in binary code (see A365139), ordered first by the number of cells, then by the value of the binary code.

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A006767 Number of 4-dimensional polyominoes with n cells.

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A290305 Number of 5-dimensional polyominoes with n cells (regarding mirror-images as distinct).

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A365143 Proper dimension of the polyomino with code A365142(n).

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