A144553 -id:A144553 - cp's OEIS frontend

A144554 Number of polyominoes with n cells whose symmetry group (excluding reflections) has order at least 2.

1, 1, 1, 3, 3, 7, 8, 25, 25, 82, 85, 302, 307, 1111, 1131, 4216, 4267, 16076, 16253, 61976, 62475, 239927, 241447, 933576, 937574, 3644073, 3653624, 14267757, 14281711, 55996279, 55968648, 220244340, 219829297, 867868410, 865120447, 3425522409, 3410557920, 13540713898, 13466370893, 53596553368

Offset: 1

Fred Schneider, Dec 28 2008

In other words, a(n) is the number of polyominoes with n cells having at least 180-degree rotational symmetry. - John Mason, Feb 14 2022

John Mason, Table of n, a(n) for n = 1..50
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
A006749 = Cases[Import["https://oeis.org/A006749/b006749.txt", "Table"], {, }][[All, 2]];
A006746 = Cases[Import["https://oeis.org/A006746/b006746.txt", "Table"], {, }][[All, 2]];
A006748 = Cases[Import["https://oeis.org/A006748/b006748.txt", "Table"], {, }][[All, 2]];
a[n_] := A000105[[n+1]] - A006749[[n]] - A006746[[n]] - A006748[[n]];
Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Aug 17 2022 *)

This is the sum of A142886, A056877, A144553, A056878 and A006747. - Joseph Myers, Dec 31 2008

a(n) = A000105(n) - A006749(n) - A006746(n) - A006748(n). - John Mason, Feb 14 2022

Edited by N. J. A. Sloane, Jan 01 2009
17 additional terms (just summing the terms from the 5 sequences specified in the description) Fred Schneider, Jan 03 2009
a(28) from John Mason, Oct 05 2021
a(29)-a(36) from John Mason, Oct 16 2021
Terms a(37) and beyond from John Mason, Feb 14 2022

A234007 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry.

Original entry on oeis.org

0, 1, 2, 9, 30, 110, 387, 1419, 5185, 19225, 71634, 269250, 1017260, 3864267, 14742260, 56470053, 217052829, 836878982

Offset: 1

John Mason, Dec 18 2013

The number of free polyominoes of size 4n that have 90-degree rotational symmetry about a point that coincides with the corner of a square, and that have not at the same time reflective symmetry. Note that for polyominoes which have a hole in the center, the center of rotation will be the corner of a square within the hole, rather than being the corner of a square of the polyomino itself. The sequence is defined for 4n rather than n as polyominoes of size not a multiple of 4 cannot have the required symmetry.

The sequence enumerates a subset of the polyominoes enumerated by A144553.

Cf. A000105, A001933, A234006, A234008, A234009, A234010, A144553.

a(8)-a(13) from Sean A. Irvine, Jul 04 2019

a(14)-a(18) from John Mason, Feb 02 2022

A324407 Number of unoriented polyomino rings of length 4n with fourfold rotational symmetry.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 21, 38, 80, 157, 336, 691, 1493, 3164, 6900, 14880, 32647, 71212, 157069, 345216, 764666, 1689978, 3756879, 8338405, 18593389, 41410352, 92583361, 206790477, 463400376, 1037575558, 2329839141, 5227759707, 11759828568, 26436550400

Offset: 1

Robert A. Russell, Feb 26 2019

Redelmeier uses these rings to enumerate polyominoes of the regular tiling {4,4} with fourfold rotational symmetry (A144553) and an even number of cells. Each cell of a polyomino ring is adjacent to (shares an edge with) exactly two other cells. For unoriented rings, a chiral ring and its congruent reflection are counted as one.
For n odd, the center of the ring is a vertex of the tiling; for n even, the center is the center of a tile.
Corrected; see A324408. - Robert A. Russell, Sep 30 2021

			For a(1)=1, the four cells form a square. For a(2)=1, the eight cells form a 3 X 3 square with the center cell omitted. For a(3)=1, the twelve cells form a 4 X 4 square with the four inner cells omitted. For a(4)=2, the sixteen cells of one ring form a 5 X 5 square with the nine inner cells omitted; the other ring is similar, but with each corner cell omitted and replaced with the cell diagonally toward the center from that corner cell.

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A324406 (oriented), A324408 (chiral), A324409 (achiral).

Cf. A144553.

a(n) = A324406(n) - A324408(n) = (A324406(n) + A324409(n)) / 2 = A324408(n) + A324409(n).

A324408 Number of chiral pairs of polyomino rings of length 4n with fourfold rotational symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 12, 29, 61, 138, 294, 649, 1402, 3073, 6696, 14676, 32199, 70764, 156062, 344209, 762433, 1687745, 3751845, 8333371, 18582147, 41399110, 92557961, 206765077, 463343343, 1037518525, 2329710014, 5227630580, 11759537552, 26436259384

Offset: 1

Robert A. Russell, Feb 26 2019

Redelmeier uses these rings to enumerate polyominoes of the regular tiling {4,4} with fourfold rotational symmetry (A144553) and an even number of cells. Each cell of a polyomino ring is adjacent to (shares an edge with) exactly two other cells. Each chiral ring is congruent to but different from its reflection; the two form a chiral pair.
These chiral rings have fourfold symmetry.
For n odd, the center of the ring is a vertex of the tiling; for n even, the center is the center of a tile.
In early September, 2021, John Mason informed me that a(16) should be 6696 instead of 6695. He supplied me with representations of all of the rings, and I slowly discovered that my program had missed one and had serious errors. After I corrected it, we did match new values for a(16), a(18), a(20), and a(22). We are reasonably confident that the values shown are now correct. - Robert A. Russell, Sep 30 2021

			For a(5) = 1, the pair is   XXX          XXX .
                            X XXX      XXX X
                           XX   X      X   XX
                           X   XX      XX   X
                           XXX X        X XXX
                             XXX        XXX

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A324406 (oriented), A324407 (unoriented), A324409 (achiral).

Cf. also A144553.

a(n) = A324406(n) - A324407(n) = (A324406(n) - A324409(n)) / 2 = A324407(n) - A324409(n).

A324409 Number of achiral polyomino rings of length 4n with fourfold rotational symmetry.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 9, 9, 19, 19, 42, 42, 91, 91, 204, 204, 448, 448, 1007, 1007, 2233, 2233, 5034, 5034, 11242, 11242, 25400, 25400, 57033, 57033, 129127, 129127, 291016, 291016

Offset: 1

Robert A. Russell, Feb 26 2019

Redelmeier uses these rings to enumerate polyominoes of the regular tiling {4,4} with fourfold rotational symmetry (A144553) and an even number of cells. Each cell of a polyomino ring is adjacent to (shares an edge with) exactly two other cells. Each achiral ring is identical to its reflection and has eightfold symmetry.
For n odd, the center of the ring is a vertex of the tiling; for n even, the center is the center of a tile.
For k > 0, the numbers of achiral rings with 8k and 8k+4 cells are the same. In the former, there are four cells in the same row or column as the center tile; we obtain the latter by moving all the cells one-half a tile away from the center in both the horizontal and vertical directions, replacing those four center-line cells with four pairs of cells.

			For a(1)=1, the four cells form a square.
For a(2)=1, the eight cells form a 3 X 3 square with the center cell omitted.
For a(3)=1, the twelve cells form a 4 X 4 square with the four inner cells omitted.
For a(4)=2, the sixteen cells of one ring form a 5 X 5 square with the nine inner cells omitted; the other ring is similar, but with each corner cell omitted and replaced with the cell diagonally toward the center from that corner cell.

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A324406 (oriented), A324407 (unoriented), A324408 (chiral).

Cf. A144553.

a(n) = 2*A324407(n) - A324406(n) = A324406(n) - 2*A324408(n) = A324407(n) - A324408(n).

A056780 Rectangular free polyominoes: number of n-celled polyominoes when the cell is a rectangle.

Original entry on oeis.org

1, 2, 3, 9, 21, 68, 208, 730, 2542, 9287, 34053, 127112, 476849, 1803636, 6851960, 26157362, 100211446, 385239872, 1485232325, 5741327939, 22246061118, 86383655207, 336093789246, 1309999171971, 5114453234510, 19998176771431, 78306018629550, 307022197845116

Offset: 1

James Sellers, Aug 28 2000

John Mason, Table of n, a(n) for n = 1..50
R. J. Mathar, Illustration of shapes up to 7-ominoes
Ed Pegg, Jr., Illustrations of polyforms
M. Vicher, Polyforms
M. Vicher, The 21 5-celled rectangular polyominoes
M. Vicher, The 21 5-celled rectangular polyominoes
Eric Weisstein's World of Mathematics, Polyrect [From _Eric W. Weisstein_, Apr 24 2009]
Wikipedia, Polyomino

Cf. A000105 (cell is square), A151522 (1-sided), A001168 (fixed).

A[s_] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, } ][[All, 2]]];
A006749 = A@006749; A006746 = A@006746; A006748 = A@006748;
A006747 = A@006747; A056877 = A@056877; A056878 = A@056878;
A144553 = A@144553; A142886 = A@142886;
a[n_] := 2*A006749[[n]] + 2*A006746[[n]] + A006748[[n]] + 2*A006747[[n]] + 2*A056877[[n]] + A056878[[n]] + A144553[[n]] + A142886[[n + 1]];
Array[a, 28] (* Jean-François Alcover, Mar 26 2020 *)

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

Edited by N. J. A. Sloane, Apr 25 2001
Two more terms from Ed Pegg Jr, May 13 2009
a(13)-a(18) from Joseph Myers, Nov 15 2010
a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A030228 Number of chiral polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 25, 88, 335, 1215, 4534, 16823, 63159, 237679, 900341, 3423201, 13073163, 50095285, 192599091, 742576616, 2870584814, 11122879867, 43191525139, 168046317330, 654998425237, 2557224396342, 9999083912711, 39153000738695, 153511081627903

Offset: 0

David W. Wilson

nonn

For n>0, A000105(n) + a(n) = A000988(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

For n>0, each chiral pair is counted as one. - Robert A. Russell, Feb 23 2022

			For a(4)=2, the two chiral tetrominoes are XXX and XX .
                                           X        XX

John Mason, Table of n, a(n) for n = 0..50 (terms 1..45 from Andrew Howroyd, 46..48 from Robert A. Russell)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A000988 (oriented), A000105 (unoriented), A030227 (achiral).

Cf. A006747, A006749, A144553 (subcategories).

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
A000988 = Cases[Import["https://oeis.org/A000988/b000988.txt", "Table"], {, }][[All, 2]];
a[n_] := A000988[[n]] - A000105[[n + 1]];
Array[a, 45] (* Jean-François Alcover, Sep 08 2019, after Graeme McRae *)

For n>0, a(n) = A000988(n) - A000105(n). - Graeme McRae, Jan 05 2006
a(n) = A006749(n) + A006747(n) + A144553(n). - Andrew Howroyd, Dec 04 2018
a(n) = A000105(n) - A030227(n). - Robert A. Russell, Feb 02 2019
For n>0, (A000988(n) - A030227(n)) / 2. - Robert A. Russell, Feb 23 2022

Terms a(23) and beyond from Andrew Howroyd, Dec 04 2018
Name edited by Robert A. Russell, Feb 03 2019
a(0)=0 corrected by John Mason, Jan 12 2023

A151522 Number of 1-sided polyrhombs with n cells.

Original entry on oeis.org

1, 2, 4, 13, 35, 120, 392, 1405, 4998, 18378, 67792, 253509, 952534, 3604624, 13699554, 52304807, 200406370, 770442286, 2970401696, 11482513428, 44491881033, 172766765654, 672186650116, 2619996250930, 10228902882021, 39996345469572, 156612023354364, 614044364443761

Offset: 1

Ed Pegg Jr, May 13 2009

Also counts 1-sided polyrects.

John Mason, Table of n, a(n) for n = 1..50
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyrhomb
Eric Weisstein's World of Mathematics, Polyrect
Wikipedia, Polyomino

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

A[n_] := With[{n6 = StringPadLeft[ToString[n], 6, "0"]}, Cases[ Import[ "https://oeis.org/A" <> n6 <> "/b" <> n6 <> ".txt", "Table"], {, }][[All, 2]]];
A006749 = A@6749; A006746 = A@6746; A006748 = A@6748; A006747 = A@6747; A056877 = A@56877; A056878 = A@56878; A144553 = A@144553; A142886 = A@142886;
a[n_] := 4*A006749[[n]] + 2*A006746[[n]] + 2*A006748[[n]] + 4*A006747[[n]] + 2*A056877[[n]] + 2*A056878[[n]] + 2*A144553[[n]] + A142886[[n + 1]];
a /@ Range[28] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = 4*A006749(n) + 2*A006746(n) + 2*A006748(n) + 4*A006747(n) + 2*A056877(n) + 2*A056878(n) + 2*A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Edited and a(13)-a(18) by Joseph Myers, Nov 24 2010

a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A324406 Number of oriented polyomino rings of length 4n with fourfold rotational symmetry.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 33, 67, 141, 295, 630, 1340, 2895, 6237, 13596, 29556, 64846, 141976, 313131, 689425, 1527099, 3377723, 7508724, 16671776, 37175536, 82809462, 185141322, 413555554, 926743719, 2075094083, 4659549155, 10455390287, 23519366120, 52872809784

Offset: 1

Robert A. Russell, Feb 26 2019

Redelmeier uses these rings to enumerate polyominoes of the regular tiling {4,4} with fourfold rotational symmetry (A144553) and an even number of cells. Each cell of a polyomino ring is adjacent to (shares an edge with) exactly two other cells. For oriented rings, chiral pairs (though congruent) are counted as two.
For n odd, the center of the ring is a vertex of the tiling; for n even, the center is the center of a tile.
Corrected; see A324408. - Robert A. Russell, Sep 30 2021

			For a(1)=1, the four cells form a square. For a(2)=1, the eight cells form a 3 X 3 square with the center cell omitted. For a(3)=1, the twelve cells form a 4 X 4 square with the four inner cells omitted. For a(4)=2, the sixteen cells of one ring form a 5 X 5 square with the nine inner cells omitted; the other ring is similar, but with each corner cell omitted and replaced with the cell diagonally toward the center from that corner cell.

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A324407 (unoriented), A324408 (chiral), A324409 (achiral).

Cf. A144553.

a(n) = A324407(n) + A324408(n) = 2*A324407(n) - A324409(n) = 2*A324408(n) + A324409(n).

A349328 Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the center of at least one square.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 9, 16, 38, 62, 147, 241, 564, 926, 2148, 3561, 8195, 13700, 31349, 52858, 120357, 204444, 463712, 792986, 1792582, 3083469, 6950579, 12018394, 27023509, 46943409, 105320716, 183715445, 411364068, 720236762, 1609836928, 2828102115

Offset: 1

John Mason, Nov 15 2021

nonn

			a(4) is 1 because of the tetromino:
   O
  OOO

John Mason, Table of n, a(n) for n = 1..50

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A349329.

For odd n, a(n) = A006746(n).

For even n, a(n) = A006746(n) - A349329(n/2).

cp's OEIS Frontend

A144554 Number of polyominoes with n cells whose symmetry group (excluding reflections) has order at least 2.

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A234007 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry.

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A324407 Number of unoriented polyomino rings of length 4n with fourfold rotational symmetry.

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A324408 Number of chiral pairs of polyomino rings of length 4n with fourfold rotational symmetry.

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A324409 Number of achiral polyomino rings of length 4n with fourfold rotational symmetry.

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A056780 Rectangular free polyominoes: number of n-celled polyominoes when the cell is a rectangle.

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A030228 Number of chiral polyominoes with n cells.

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A151522 Number of 1-sided polyrhombs with n cells.

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A324406 Number of oriented polyomino rings of length 4n with fourfold rotational symmetry.