A351142 -id:A351142 - cp's OEIS frontend

A144553 Number of chiral pairs of polyominoes with n cells that have precisely the symmetry group of order 4 generated by 90-degree rotations.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 12, 7, 0, 0, 44, 25, 0, 0, 165, 90, 0, 0, 603, 319, 0, 0, 2235, 1136, 0, 0, 8283, 4088, 0, 0, 30936, 14868, 0, 0, 116111, 54526, 0, 0, 438465, 201527, 0, 0, 1663720, 750169, 0, 0, 6342211, 2809931, 0, 0, 24273767

Offset: 1

N. J. A. Sloane, Jan 01 2009

nonn

The values for n>28 were produced by a set of programs, the most difficult of which is attached. There is no guarantee that the values are correct, although presumably Shirakawa has calculated them through a(45). The attached program can be altered to count only achiral polyominoes, and those results match those of A142886, which uses a very different method. The difficulties lie in determining each inner loop (A324408 and A324409) and in determining connections within the inner loop (bad_connection subroutine). The last bug I found in the program affected only polyominoes with 72 or more cells. - Robert A. Russell, May 23 2020
These are polyominoes of the regular tiling with Schläfli symbol {4,4}. In late August, 2021, John Mason informed me that there were errors for a(44) and higher. My error in a(44) was a copying error, but later entries were wrong because of my programming errors. After making corrections (see attached C++ program), our values now match. John uses a unique calculation of his own devising. Since it is quite different from Redelmeier's inner rings, the match gives us some confidence in the current values. - Robert A. Russell, Nov 01 2021
Polyominoes with precisely 90-degree symmetry centered about square centers and vertices are enumerated by A351142 and A234007 respectively. - John Mason, Feb 17 2022

			For a(8)=1, the polyomino is a central 2 X 2 square with one cell attached to each edge of that square. - _Robert A. Russell_, Nov 01 2021

Robert A. Russell, Table of n, a(n) for n = 1..95
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...
Robert A. Russell, C++ Program
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351142, A234007.
Cf. A324408, A324409 (inner rings).
Cf. A348848 (C90), A348849 (F90).

A006747 = Cases[Import["https://oeis.org/A006747/b006747.txt",
     "Table"], {, }][[All, 2]];
A006749 = Cases[Import["https://oeis.org/A006749/b006749.txt",
     "Table"], {, }][[All, 2]];
A030228 = Cases[Import["https://oeis.org/A030228/b030228.txt",
     "Table"], {, }][[All, 2]];
a[n_] := A030228[[n+1]] - A006747[[n]] - A006749[[n]];
Array[a, 43] (* Jean-François Alcover, Sep 09 2019, updated Aug 17 2022 *)

a(n) = A030228(n) - A006747(n) - A006749(n). - Jean-François Alcover, Sep 09 2019, after Andrew Howroyd in A030228.
a(n) = (A348848(n/4)+A348849(n)-A142886(n)) / 2, where the first two are F90 and C90 of the Shirakawa link. - Robert A. Russell, Nov 01 2021
a(n) = A351142(n) + A234007(n/4) if n is a multiple of 4, otherwise a(n) = A351142(n). - John Mason, Feb 17 2022

a(28) added by Andrew Howroyd, Dec 04 2018
a(29)-a(91) added by Robert A. Russell, May 23 2020
Warning: It seems that the C++ program and the Mathematica program produce different results. This means that the b-file, and possibly even the terms in the DATA lines, are suspect. - N. J. A. Sloane, Aug 17 2022
After John Mason's Apr 15 2023 correction to the b-file of A006749, the discrepancy disappeared. - Andrey Zabolotskiy, Jan 18 2024

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.

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A144553 Number of chiral pairs of polyominoes with n cells that have precisely the symmetry group of order 4 generated by 90-degree rotations.

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