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

Author

N. J. A. Sloane, Jan 01 2009

Keywords

• nonn

Comments

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

Examples

			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
		

Links

• 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.

Crossrefs

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).

Programs

Mathematica

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 *)

Formula

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

Extensions

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