A234010 -id:A234010 - cp's OEIS frontend

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

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

A234008 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry.

Original entry on oeis.org

0, 1, 4, 16, 60, 231, 877, 3362, 12905, 49825, 193003, 750581, 2927792, 11453171, 44911853, 176499605, 694954416, 2741031257, 10827727980, 42831355495, 169640762209, 672657218163, 2669990735153, 10608176066076, 42184579054003

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, and that have not at the same time 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.

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

Cf. A000105, A001933, A234006, A234007, A234009, A234010, A006747.

a(12)-a(18) from John Mason, Dec 13 2021

a(19)-a(25) from John Mason, Apr 15 2023

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

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

cp's OEIS Frontend

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

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A234008 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no 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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A234009 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner.

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