A346799 -id:A346799 - cp's OEIS frontend

A056877 Number of polyominoes with n cells, symmetric about two orthogonal axes.

0, 1, 1, 1, 1, 2, 3, 4, 4, 8, 10, 15, 17, 30, 35, 60, 64, 117, 128, 236, 241, 459, 476, 937, 912, 1813, 1789, 3706, 3456, 7187, 6779, 14712, 13161, 28571, 25839, 58457, 50348, 113798, 98957, 232718, 193375, 453969, 380522, 927601, 745248, 1813219, 1468202, 3702063

Offset: 1

N. J. A. Sloane, Sep 03 2000

nonn

This sequence counts polyominoes with exactly the symmetry group D_4 generated by horizontal and vertical reflections.
The subset of (2n)-ominoes with edge centers in this set are enumerated by A346799(n). - Robert A. Russell, Dec 15 2021
Polyominoes centered about square centers and vertices are enumerated by A351190 and A351191 respectively. - John Mason, Feb 15 2022

			For a(8)=4, the four octominoes with exactly fourfold symmetry and axes of symmetry parallel to the edges of the cells are a row of eight cells, two adjacent rows of four cells, a row of four cells with another four cells adjacent to its center cells, and a row of four cells with another four cells adjacent to its end cells (but not in the original row):
  XXXXXXXX
.
   XXXX
   XXXX
.
   XX
  XXXX
   XX
.
  X  X
  XXXX
  X  X

Robert A. Russell, Table of n, a(n) for n = 1..81
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...
Wikipedia, Symmetries of polyominoes

Cf. A000105, A001168, A006746, A006748, A056878, A006747, A006749, A346799, A351190, A351191.

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

a(n) = A351190(n) + A346799(n/2) + A351191(n/4) if we accept the convention that Axxxxxx(y) = 0 for any noninteger y. - John Mason, Feb 15 2022

More terms from Robert A. Russell, Jan 16 2019

A030227 Number of achiral polyominoes with n cells.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 34, 70, 121, 250, 441, 912, 1630, 3375, 6092, 12624, 22961, 47616, 87136, 180811, 332549, 690398, 1275166, 2648422, 4909364, 10199792, 18966700, 39416488, 73497642, 152777230, 285569898, 593717419, 1112188817, 2312672439, 4340728280

Offset: 0

David W. Wilson

nonn

Polyominoes with n cells and at least one line of reflection symmetry. - Joshua Zucker, Mar 08 2008

This sequence can most readily be calculated by enumerating fixed polyominoes for three different axes of symmetry: 1) a line composed of the diagonals of cells, A346800, 2) a line composed of edges of cells, and 3) a line composed of lines connecting the centers of adjacent cells, A346799. For the second case, any fixed polyomino just touching the edge line is reflected on the other side, so that sequence is A001168(n/2) for even values of n and zero otherwise. These three sequences together include each achiral polyomino exactly twice. - Robert A. Russell, Aug 04 2021

			For a(4)=3, the achiral tetrominoes are a 2 X 2 square, a 1 X 4 rectangle, and a cell plus three cells adjacent to it (forming a shortened T).

John Mason, Table of n, a(n) for n = 0..50 (terms 1..48 from Robert A. Russell.)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A000988 (oriented), A000105 (unoriented), A030228 (chiral).
Cf. A006746, A006748, A056877, A056878, A142886 (subcategories).
Cf. A001168, A346799, A346800.

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_] := 2*A000105[[n + 1]] - A000988[[n]];
Array[a, 45] (* Jean-François Alcover, Sep 08 2019, after Andrew Howroyd *)

a(n) = A000105(n) - A030228(n) = 2*A000105(n) - A000988(n). - Andrew Howroyd, Dec 04 2018
a(n) = A006746(n) + A006748(n) + A056877(n) + A056878(n) + A142886(n) = A000988(n) - 2*A030228(n). - Robert A. Russell, Feb 02 2019
For odd n, a(n) = (A346799(n) + A346800(n)) / 2; for even n, a(n) = (A346799(n) + A001168(n/2) + A346800(n)) / 2. - Robert A. Russell, Aug 04 2021

a(23)-a(36) from Andrew Howroyd, Dec 04 2018
Name edited by Robert A. Russell, Feb 03 2019
Offset changed to 0, and a(0) added by John Mason, Jan 12 2023

A346800 Number of fixed polyominoes with n cells that have a diagonal axis of symmetry going from lower left to upper right.

Original entry on oeis.org

1, 0, 2, 1, 5, 4, 16, 13, 54, 46, 186, 167, 660, 612, 2384, 2267, 8726, 8464, 32278, 31822, 120419, 120338, 452420, 457320, 1709845, 1745438, 6494848, 6686929, 24779026, 25703792, 94899470, 99096382, 364680344

Offset: 1

Robert A. Russell, Aug 04 2021

nonn

This is one of three sequences needed to calculate the number of achiral polyominoes, A030227. The three sequences together contain exactly two copies of each achiral polyomino. This is the DL sequence in the Shirakawa link. The sequence can be calculated using Redelmeier's method; one chooses an original cell such that no cells in its LL-UR diagonal on one side of it are eligible, nor are any cells in lower LL-UR diagonals. Cells in that original diagonal are counted as one; all others count as two. Jensen's transfer matrix method (see Knuth POLYNUM program) could likely be modified to enumerate this sequence for many more terms; instead of rows, one uses diagonals.

The sequence also enumerates free polyominoes of size 4*n with maximal symmetry that have a center of rotation on a vertex of the underlying square matrix, which are a subset of those enumerated by A142886. - John Mason Jan 27 2022

			For a(5)=5, the polyominoes are:  XXX   X     X     XX     X
                                    X   X     XX     XX   XXX
                                    X   XXX    XX     X    X

John Mason, Table of n, a(n) for n = 1..50 (terms 1..47 from Robert A. Russell).
D. E. Knuth, Program
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A030227, A346799, A006748, A056878, A142886.

a(n) = 2*A006748(n) + 2*A056878(n) + A142886(n). - John Mason Jan 27 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

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

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

A056877 Number of polyominoes with n cells, symmetric about two orthogonal axes.

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A030227 Number of achiral polyominoes with n cells.

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A346800 Number of fixed polyominoes with n cells that have a diagonal axis of symmetry going from lower left to upper right.

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

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

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