

A144553


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


30



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
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OFFSET

1,12


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 90degree symmetry centered about square centers and vertices are enumerated by A351142 and A234007 respectively.  John Mason, Feb 17 2022


LINKS

D. H. Redelmeier, Table 3 of Counting polyominoes...


FORMULA



EXAMPLE

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


MATHEMATICA

"Table"], {_, _}][[All, 2]];
"Table"], {_, _}][[All, 2]];
"Table"], {_, _}][[All, 2]];


CROSSREFS

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351142, A234007.


KEYWORD

nonn,obsc


AUTHOR



EXTENSIONS

Warning: It seems that the C++ program and the Mathematica program produce different results. This means that the bfile, and possibly even the terms in the DATA lines, are suspect. That is why I added the keyword "obsc".  N. J. A. Sloane, Aug 17 2022


STATUS

approved



