A351127 -id:A351127 - cp's OEIS frontend

A142886 Number of polyominoes with n cells that have the symmetry group D_8.

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 12, 7, 0, 0, 20, 11, 0, 0, 45, 20, 0, 0, 80, 36, 0, 0, 173, 65, 0, 0, 310, 117, 0, 0, 664, 216, 0, 0, 1210, 396, 0, 0, 2570, 736, 0, 0, 4728, 1369, 0, 0, 9976, 2558, 0, 0, 18468, 4787, 0, 0, 38840

Offset: 0

N. J. A. Sloane, Jan 01 2009

nonn

This is the largest possible symmetry group that a polyomino can have.

Polyominoes with such symmetry centered about square centers and vertices are enumerated by A351127 and A346800 respectively. - John Mason, Feb 16 2022

			The monomino has eight-fold symmetry. The tetromino with eight-fold symmetry is four cells in a square. The pentomino with eight-fold symmetry is a cell and its four adjacent cells.

Robert A. Russell, Table of n, a(n) for n = 0..163
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...
Index entries for sequences related to groups

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351127, A346800.

Cf. A376971 (polycubes with full symmetry).

a(n) = A351127(n) + A346800(n/4) if n is a multiple of 4, otherwise a(n) = A351127(n). - John Mason, Feb 16 2022

Name corrected by Wesley Prosser, Sep 06 2017
a(28) added by Andrew Howroyd, Dec 04 2018
More terms from Robert A. Russell, Jan 13 2019

A346799 Number of fixed polyominoes with n cells that have a horizontal axis of symmetry that passes through the centers of cells.

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 24, 36, 86, 133, 314, 499, 1164, 1888, 4366, 7192, 16522, 27548, 62954, 106004, 241203, 409492, 928376, 1587151, 3586999, 6169400, 13904736, 24041597, 54053950, 93896826, 210654990, 367450477, 822754494

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 FL sequence in the Shirakawa link. The sequence can be quickly calculated using Redelmeier's method; each polyomino cell in the lowest row is counted as one, while all the other polyomino cells are counted as two. Jensen's transfer matrix method (see Knuth POLYNUM program) could be modified to enumerate this sequence for over 100 terms; one needs only to keep track of the number of polyomino cells in the original row.

John Mason has pointed out that a(n) is also the number of achiral (2n)-ominoes with twofold rotational symmetry centered at the center of an edge. Just add to each polyomino its reflection in its leftmost edge to obtain these, the subset of A056877 with edge centers. - Robert A. Russell, Dec 15 2021

			For a(5)=7, the polyominoes are:    X
X       X   XX   XX    X            X
XXX   XXX   X     X   XXX   XXXXX   X
X       X   XX   XX    X            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, A056877, A346800.

a(n) = A351127(n) + 2 * A351190(n) + A346799(n / 2) + 2 * A349328(n), setting A346799(n / 2) = 0 for noninteger arguments. - John Mason, Mar 13 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.

A377334 Number of n-celled polycubes with full symmetry and the rotation point of the symmetries in the center of a cell (that may or may not be part of the polycube).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 3, 1, 1, 0, 0, 4, 5, 4, 1, 0, 0, 5, 7, 4, 3, 0, 0, 8, 10, 6, 3, 0, 0, 12, 14, 8, 5, 0, 0, 22, 21, 21, 7, 0, 0, 32, 32, 20, 12, 2, 0, 50, 48, 36, 16, 1, 1

Offset: 1

Pontus von Brömssen, Oct 25 2024

nonn

Index entries for sequences related to polyominoes.

Cf. A351127, A376971, A377335.

a(n) = A376971(n) if n is not divisible by 8, otherwise a(n) = A376971(n)-A377335(n/8).

Conjecture: For n >= 62, a(n) > a(n-1) if and only if n is a multiple of 6.

cp's OEIS Frontend

A142886 Number of polyominoes with n cells that have the symmetry group D_8.

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A346799 Number of fixed polyominoes with n cells that have a horizontal axis of symmetry that passes through the centers of cells.

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

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A377334 Number of n-celled polycubes with full symmetry and the rotation point of the symmetries in the center of a cell (that may or may not be part of the polycube).

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