A000104 -id:A000104 - cp's OEIS frontend

A103468 Number of polyominoes consisting of n regular unit 10-gons.

Original entry on oeis.org

1, 1, 4, 19, 127, 985, 8350, 73675, 664411, 6078768, 56198759, 523924389, 4918127659

Offset: 1

Sascha Kurz, Feb 07 2005

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, arXiv:math.CO/0605144
S. Kurz, k-polyominoes.

Cf. A000577, A000104, A000228, A103465 - A103473, A115071, A120102 - A120104.

More terms from Sascha Kurz, Jun 09 2006

A120103 Number of polyominoes consisting of 9 regular unit n-gons.

Original entry on oeis.org

160, 1285, 14445, 6572, 14982, 65323, 280014, 664411, 1908239, 1314914, 1968684, 4158216, 9707046, 17054708, 33522023, 26019735, 33942901, 56537856, 100952307, 153177526, 251530341, 208524646, 254079408, 374310135, 586169115, 812395658

Offset: 3

Sascha Kurz, Jun 09 2006

nonn

			a(3)=160 because there are 160 polyiamonds consisting of 9 triangles and a(4)=1285 because there are 1285 polyominoes consisting of 9 squares.

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, preprint, arXiv:math.CO/0605144

Cf. A000577, A000104, A000105, A000228, A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A103465, A120102, A120104.

A268371 Triangle read by rows: T(n,k) is the number of free polyominoes with width n and height 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 6, 24, 1, 12, 181, 1051, 1, 30, 941, 21992, 238048, 1, 65, 4662, 228013, 9735647, 195284973, 1, 158, 23691, 2337694, 195616247, 15256715219, 577169894573

Offset: 1

John Mason, Feb 03 2016

Superset of Craig Knecht's A268311.

			The first rows of the array are:
1;
1, 2;
1, 6, 24;
1, 12, 181, 1051;
...
T(4,3)=181 = 15+39+59+42+21+4+1: In the 4x3 square fit 15 6-ominoes, 39 7-ominoes, 59 8-ominoes, 42 9-ominoes, 21 10-ominoes, 4 11-ominoes or 1 12-omino. - _R. J. Mathar_, Jun 07 2020

Jean-Luc Manguin, Table of n, a(n) for n = 1..28
R. J. Mathar, Corrigendum to "Polyomino Enumeration Results (Parkin et al, SIAM Fall Meeting 1967)" viXra:1905.0474 (2019)
R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy) page 9 (incorrect at n=15).
Index entries for sequences related to polyominoes

Cf. A268311 (right diagonal), A335711 (column n=2), A000105 (free polyominoes), A292357 (equival. for fixed polys).

a(16)-a(28) from Jean-Luc Manguin, May 25 2020

A103466 Number of polyominoes consisting of n regular unit octagons.

Original entry on oeis.org

1, 1, 3, 11, 50, 269, 1605, 10102, 65323, 430302, 2868320, 19299334, 130807068, 892075515, 6115673262

Offset: 1

Sascha Kurz, Feb 07 2005

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, arXiv:math/0605144 [math.CO], 2006.
S. Kurz, k-polyominoes.

Cf. A000577, A000104, A000228, A103465, A103465, A103467, A103468.

Cf. A000577, A000104, A000228, A103465, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A115071, A120102, A120103, A120104.

More terms from Sascha Kurz, Jun 09 2006

Definition corrected by John Mason and Sascha Kurz, Sep 20 2020

A103467 Number of polyominoes consisting of n regular unit 9-gons.

Original entry on oeis.org

1, 1, 3, 14, 82, 585, 4418, 34838, 280014, 2285047, 18838395, 156644526, 1311575691

Offset: 1

Sascha Kurz, Feb 07 2005

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, arXiv:math/0605144 [math.CO], 2006.
S. Kurz, k-polyominoes.

Cf. A000577, A000104, A000228, A103465, A103466, A103468, A103469, A103470, A103471, A103472, A103473, A115071, A120102, A120103, A120104.

More terms from Sascha Kurz, Jun 09 2006

Definition corrected by John Mason and Sascha Kurz, Sep 20 2020

A054361 Number of polyominoes with n cells without holes that do not tile the plane.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 20, 198, 1390, 9474, 35488, 178448, 696371, 2721544, 10683110, 41334494, 155723774, 596182769, 2257379379, 8587521496, 32688629235, 124568505590, 475147925759, 1815832051949

Offset: 1

Joe Keane (jgk(AT)jgk.org)

			a(7) = 3.

Posting by Glenn C. Rhoads (rhoads(AT)paul.rutgers.edu) to rec.puzzles, Feb 17 2000.

Craig S. Kaplan, Heesch Numbers of Unmarked Polyforms
Craig S. Kaplan, Heesch Numbers of Unmarked Polyforms, arXiv:2105.09438 [cs.CG], 2021. See Table 1 and Table 2.
Joseph Myers, Polyomino tiling

Cf. A000104, A054360, A070768, A071334.

Corrected and extended by Joseph Myers, May 05 2002
More terms from Joseph Myers, Nov 04 2003
a(24) and a(25) from Joseph Myers, Nov 17 2010

A089458 Number of n-celled polyominoes with exactly 6 holes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 2131, 38022, 480713, 4872679, 42360239

Offset: 1

Joseph Myers, Nov 04 2003

Tomás Oliveira e Silva, Animal enumerations on the {4,4} Euclidean tiling

Cf. A001419, A000104, A057418, A089454, A089455, A089456, A089457.

a(27)-a(28) from Tomás Oliveira e Silva's page added by Andrey Zabolotskiy, Oct 02 2022

A259088 Number of n-celled polyominoes which are of square type.

Original entry on oeis.org

1, 0, 1, 1, 6, 7, 25, 80, 255, 795, 2919, 10378, 36590, 130716, 480773, 1774714, 6543633, 24208803, 90331307, 339178346, 1277308160, 4819109576

Offset: 1

N. J. A. Sloane, Jun 18 2015

			a(7) = 25 with 7 free 7-ominoes with a 3x3 hull and 18 with 4x4 hull. a(8) = 80 with 3 free 8-ominoes with a 3x3 hull and 77 with a 4x4 hull. a(9) = 255 with 1 free 9-omino in a 3x3 hull, 181 in a 4x4 hull and 73 in a 5x5 hull. - _R. J. Mathar_, May 24 2019

R. J. Mathar, Corrigendum to "Polynomino enumeration results (Parkin et al, SIAM Fall Meeting 1967)", vixra:1905.0474
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy). See page 21. The 15th entry is not included because their value for A000105(15) was incorrect.

Cf. A259087, A268311, A000105.

a(n)+A259087(n) = A000105(n).

a(15)-a(16) from R. J. Mathar, May 22 2019
a(17) from R. J. Mathar, May 27 2019
a(18)-a(22) from John Mason, Nov 19 2021

A342537 Number of free polyominoes with n cells having simply-connected interiors.

Original entry on oeis.org

1, 1, 2, 5, 12, 35, 108, 368, 1283, 4639, 17003, 63222, 236819, 893515, 3387953, 12904855, 49335965, 189243115

Offset: 1

John Mason, Allan C. Wechsler, Mar 15 2021

The polyominoes referred to in the definition are those having a single traceable perimeter. The definition therefore includes the heptomino with a hole, as in this case the border, although not self-avoiding (it meets itself at a corner) is still traceable; it excludes the square octomino with a hole, whose border consists of two distinct polygons. Thus starting from n=8, A000104(n) < a(n) < A000105(n).

Cf. A000104 (free polyominoes without holes), A000105 (free polyominoes).

a(17)-a(18) from John Mason, Jan 04 2023

A056880 Number of n-celled polyominoes without holes, symmetric about axis 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 4, 4, 6, 10, 11, 16, 18, 33, 34, 56, 56, 110, 106, 193, 177, 372, 335, 662, 571, 1269, 1084, 2282, 1875, 4365, 3568, 7914, 6257, 15141, 11935, 27615, 21170, 52911, 40486, 96980, 72490, 186144, 138981, 342638, 250814, 658884, 482029, 1217327, 875697, 2345272, 1686803

Offset: 1

N. J. A. Sloane, Sep 03 2000

nonn

In other words, free polyominoes, without holes, with both vertical and horizontal symmetries but no other. - John Mason, Oct 10 2022

Tomás Oliveira e Silva, Enumeration of polyominoes

Cf. A000104, A056877 (with holes allowed).

a(29)-a(52) from John Mason, Oct 10 2022

cp's OEIS Frontend

A103468 Number of polyominoes consisting of n regular unit 10-gons.

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A120103 Number of polyominoes consisting of 9 regular unit n-gons.

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A268371 Triangle read by rows: T(n,k) is the number of free polyominoes with width n and height 1<=k<=n.

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A103466 Number of polyominoes consisting of n regular unit octagons.

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A103467 Number of polyominoes consisting of n regular unit 9-gons.

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A054361 Number of polyominoes with n cells without holes that do not tile the plane.

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A089458 Number of n-celled polyominoes with exactly 6 holes.

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A259088 Number of n-celled polyominoes which are of square type.

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A342537 Number of free polyominoes with n cells having simply-connected interiors.

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A056880 Number of n-celled polyominoes without holes, symmetric about axis 2.

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