A234008 -id:A234008 - cp's OEIS frontend

A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

0, 0, 0, 1, 1, 5, 4, 18, 19, 73, 73, 278, 283, 1076, 1090, 4125, 4183, 15939, 16105, 61628, 62170, 239388, 240907, 932230, 936447, 3641945, 3651618, 14262540, 14277519, 55987858, 55961118, 220223982, 219813564, 867835023, 865091976, 3425442681

Offset: 1

N. J. A. Sloane

nonn

This sequence gives the number of free polyominoes with symmetry group "R" in Redelmeier's notation. See his Tables 1 and 3, also the column "Rot" in Oliveira e Silva's table.

The rotation center of a polyomino with this symmetry may lie at the center of a square, the middle of an edge, or a vertex of a square. These subsets are enumerated by A351615, A234008 and A351616 respectively. - John Mason, Feb 17 2022, reformulated by Günter Rote, Oct 19 2023

			a(2) = 0 because the "domino" polyomino has symmetry group of order 4.
For n=3, the three-celled polyomino [ | | ] has group of order 4, and the polyomino
. [ ]
. [ | ]
has only reflective symmetry, so a(3) = 0.
a(4) = 1 because of (in Golomb's notation) the "skew tetromino".

S. W. Golomb, Polyominoes, Princeton Univ. Press, NJ, 1994.
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
Tomás Oliveira e Silva, Enumeration of polyominoes
Tomás Oliveira e Silva, Numbers of polyominoes classified according to Redelmeier's symmetry classes (an extract from the previous link)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.
Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351615, A234008, A351616.
Polyomino rings of length 2n with twofold rotational symmetry: A348402, A348403, A348404.

a(n) = A351615(n) + A234008(n/2) + A351616(n/2) for even n, otherwise a(n) = A351615(n). - John Mason, Feb 17 2022

Extended to n=28 by Tomás Oliveira e Silva
a(1)-a(3) prepended by Andrew Howroyd, Dec 04 2018
Edited by N. J. A. Sloane, Nov 28 2020
a(29)-a(36) from John Mason, Oct 16 2021

A001933 Number of chessboard polyominoes with n squares.

Original entry on oeis.org

2, 1, 4, 7, 24, 62, 216, 710, 2570, 9215, 34146, 126853, 477182, 1802673, 6853152, 26153758, 100215818, 385226201, 1485248464, 5741275753, 22246121356, 86383454582, 336094015456, 1309998396933, 5114454089528, 19998173763831, 78306021876974, 307022186132259, 1205243906123956, 4736694016531135

Offset: 1

N. J. A. Sloane

Chessboard-colored polyominoes, considering to be distinct two shapes that cannot be mapped onto each other by any form of symmetry. For example, there are two distinct monominoes, one black, one white. There is only one domino, with one black square, and one white. - John Mason, Nov 25 2013

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 = 1..50
Joseph Myers, Chessboard polyominoes

Cf. A001071, A000105, A121198, A234006 (free polyominoes of size 2n that have at least reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares), A234007 (free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry), A234008 (free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry).

For odd n, a(n) = 2*A000105(n).
For n multiple of 2 but not of 4, a(n) = 2*A000105(n) - (A234006(n/2) + A234008(n/2)).
For n multiple of 4, a(n) = 2*A000105(n) - (A234006(n/2) + A234008(n/2) + A234007(n/4)). - John Mason, Dec 23 2021

a(14)-a(17) from Joseph Myers, Oct 01 2011
a(18)-a(23) from John Mason, Dec 05 2013
a(24)-a(30) from John Mason, Dec 23 2021

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

A234006 Free polyominoes with 2n squares, having reflectional symmetry on axis that coincides with edges.

Original entry on oeis.org

1, 2, 4, 11, 35, 114, 392, 1381, 4998, 18292, 67791, 253182, 952527, 3603389, 13699516, 52300071, 200406183, 770424072, 2970400815, 11482442855, 44491876993, 172766491178, 672186631950, 2619995178793, 10228902801505, 39996341268584, 156612023001490, 614044347934591

Offset: 1

John Mason, Dec 18 2013

nonn

The number of free polyominoes of size 2n that have reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.

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

Cf. A000105, A001168, A001933, A151525, A182645, A234007, A234008, A234009, A234010, A349329, A346799, A346800, A351191.

A151525 = Cases[Import["https://oeis.org/A151525/b151525.txt", "Table"], {, }][[All, 2]];
A182645 = Cases[Import["https://oeis.org/A182645/b182645.txt", "Table"], {, }][[All, 2]];
A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {, }][[All, 2]];
a[n_] := If[OddQ[n], A151525[[n]], A151525[[n]] + A182645[[n/2]] - A001168[[n/2]]];
Array[a, 28] (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)

a(2*n+1) = A151525(2*n+1), a(2*n) = A151525(2*n) + A182645(n) - A001168(n). - Andrew Howroyd, Dec 05 2018

If n odd, a(n) = A349329(n) + A346799(n), otherwise a(n) = A349329(n) + A346799(n) + A346800(n/2) + A351191(n/2). - John Mason, Mar 15 2023

a(12)-a(28) from Andrew Howroyd, Dec 05 2018

A121198 Number of one-sided chessboard polyominoes with n cells (similar to but different from A001071).

Original entry on oeis.org

2, 1, 4, 10, 36, 110, 392, 1371, 5000, 18251, 67792, 253040, 952540, 3602846, 13699554, 52298057, 200406388, 770416390, 2970401696, 11482413680, 44491881090, 172766379334, 672186650116, 2619994749395, 10228902882212, 39996339612824, 156612023354364, 614044341535992

Offset: 1

N. J. A. Sloane, Aug 17 2006

Consider the tiling of the plane with squares of two different sizes as seen for example in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. Sequence gives the number of "n-PairSquares", that is, polyominoes or animals that can be formed on this tiling from "n big or little squares, where the conjunction between two squares must involve an entire edge at least". - Original description (N. J. A. Sloane, Aug 17 2006, with quote from Livio Zucca's site)

Also counts one-sided polyominoes cut from an infinite chessboard with the usual coloring (big and little squares in Fig. 2.4.2(g) of Grünbaum and Shephard are equivalent to the two colors on a chessboard, and ignoring connections that are not a whole edge of one square means the connectivity is also equivalent); see Myers link regarding difference from A001071 for even terms a(6) onwards. - Joseph Myers, Oct 01 2011

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

John Mason, Table of n, a(n) for n = 1..50
Joseph Myers, Chessboard polyominoes
Livio Zucca, PolyMultiForms

Cf. A001071, A001933, A121195, A121196, A000105 (free polyominoes), A030228 (chiral polyominoes), A234009 (free polyominoes with 90-degree rotational symmetry about a square corner), A234007 (chiral polyominoes with 90-degree rotational symmetry about a square corner), A346799 (achiral polyominoes with twofold rotational symmetry around the center of an edge), A234008 (chiral polyominoes with 180-degree rotational symmetry about the center of an edge).

From John Mason, Dec 24 2021: (Start)
For odd n, a(n) = 2*A000105(n) + 2*A030228(n).
For n multiple of 2 but not of 4, a(n) = 2*A000105(n) + 2*A030228(n) - A346799(n/2) - 2*A234008(n/2).
For n multiple of 4, a(n) = 2*A000105(n) + 2*A030228(n) - A346799(n/2) - 2*A234008(n/2) - A234009(n/4) - A234007(n/4). (End)

a(6)-a(17) by Joseph Myers, Oct 01 2011
a(18)-a(21) by John Mason, Jan 04 2014
Erroneous a(21) removed by John Mason, Feb 12 2021
a(21)-a(28) from John Mason, Dec 24 2021

A234009 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner.

Original entry on oeis.org

1, 1, 4, 10, 35, 114, 403, 1432, 5239, 19271, 71820, 269417, 1017920, 3864879

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, independently of any other symmetries. 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 multiple of 4 cannot have the required symmetry.

Cf. A000105, A001933, A234006, A234007, A234008, A234010.

a(8)-a(14) from John Mason, Dec 13 2021

A234010 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side.

Original entry on oeis.org

1, 2, 6, 19, 67, 241, 901, 3398, 12991, 49958, 193317, 751080, 2928956, 11455059, 44916219, 176506797, 694970938, 2741058805, 10827790934, 42831461499, 169641003412, 672657627655, 2669991663529, 10608177653227, 42184582641002

Offset: 1

John Mason, Dec 18 2013

The number of free polyominoes of size 2n that have 180-degree rotational symmetry about a point that coincides with the midpoint of a side a square, independently of any reflective symmetry. Note that for polyominoes which have a hole in the center, the center of rotation will be the midpoint of a side of a square within the hole, rather than being the midpoint of a side of a square of the polyomino itself. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.

Cf. A000105, A001933, A234006, A234007, A234008, A234009, A346799.

a(n) = A346799(n) + A234008(n).

More terms from John Mason, Dec 17 2021

More terms from John Mason, Apr 15 2023

A351615 Number of free polyominoes with n squares, having 180-degree rotational symmetry about the center of a square, and no other symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 4, 0, 19, 1, 73, 4, 283, 25, 1090, 106, 4183, 463, 16105, 1892, 62170, 7752, 240907, 31212, 936447, 125609, 3651618, 503165, 14277519, 2014826, 55961118, 8058790, 219813564, 32231897, 865091976, 128897247, 3410498446, 515554767, 13466262587, 2062445063, 53245595410, 8252868076, 210800433323

Offset: 1

John Mason, Feb 15 2022

nonn

The sequence enumerates a subset of the polyominoes enumerated by A006747.

			a(5)=1 because of:
   OO
   O
  OO

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

Cf. A006747, A234008, A351616.

More terms from John Mason, Apr 15 2023

A351616 Number of free polyominoes with 2n squares, having 180-degree rotational symmetry about a vertex of a square, and no other symmetry.

Original entry on oeis.org

0, 0, 1, 2, 12, 43, 174, 657, 2571, 9911, 38633, 150437, 588544, 2306204, 9061179, 35665587, 140648710, 555514177, 2197298561, 8702439446, 34506900216, 136972867380, 544237912916, 2164377093376, 8614630304437

Offset: 1

John Mason, Feb 15 2022

The sequence enumerates a subset of the polyominoes enumerated by A006747.

			a(3)=1 because of:
   OOO
  OOO

Cf. A006747, A234008, A351615.

a(17)-a(25) from John Mason, Feb 26 2023

A361625 Number of free polyominoes with checkerboard-pattern-colored vertices with n cells.

Original entry on oeis.org

1, 1, 3, 7, 20, 60, 204, 702, 2526, 9180, 33989, 126713, 476597, 1802109, 6850969, 26151529, 100207548, 385217382, 1485216987, 5741240989, 22246000726, 86383317470, 336093551268, 1309997856337, 5114452295933, 19998171631076, 78306014924606, 307022177714062

Offset: 1

Andrey Zabolotskiy, Mar 19 2023; thanks to John Mason for his help

nonn

Also, number of polysticks of size n (see A019988), with the requirement that any two sticks are connected by a sequence of adjacent, alternately horizontal and vertical sticks. - Pontus von Brömssen, Sep 01 2023

			There are 2 ways to color the 4 corners of a monomino with black and white colors alternatingly, but they are related by a rotation or a reflection, so a(1) = 1. a(2) is also 1 because the two ways to color the 6 vertices of a domino with black and white colors in the checkerboard pattern are related to each other by a reflection or a rotation. The same is true for the stick tromino, but the two ways to color the 8 vertices of the L-tromino are inequivalent, so a(3) = 3.
For n = 3, the a(3) = 3 allowed polysticks are:
  _     _
  _|  _|   _|_

Andrey Zabolotskiy, Table of n, a(n) for n = 1..50
Index entries for sequences related to polyominoes.

A122675 is the 3-dimensional analog based on polycubes.
Cf. A001933, A019988, A359689.
Cf. A000105, A351190, A351142, A351127, A349328, A346799, A234008.
5th row of A366766.

a(n) = 2 * A000105(n) - (A351190(n) + A351142(n) + A351127(n) + A349328(n) + A346799(n/2) + A234008(n/2)), where the last two terms are only included if 2|n. I.e., every free polyomino is counted twice here unless it is symmetric with respect to a Pi/2 rotation centered at a cell, or a Pi rotation centered at an edge, or a reflection with respect to an axis parallel to the grid and passing through cells.

cp's OEIS Frontend

A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

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A001933 Number of chessboard polyominoes with n squares.

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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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A234006 Free polyominoes with 2n squares, having reflectional symmetry on axis that coincides with edges.

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A121198 Number of one-sided chessboard polyominoes with n cells (similar to but different from A001071).

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A234009 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner.

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A234010 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side.

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A351615 Number of free polyominoes with n squares, having 180-degree rotational symmetry about the center of a square, and no other symmetry.

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A351616 Number of free polyominoes with 2n squares, having 180-degree rotational symmetry about a vertex of a square, and no other symmetry.

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