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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.

%I A144553 #57 Jan 18 2024 11:36:47
%S A144553 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,
%T A144553 0,0,2235,1136,0,0,8283,4088,0,0,30936,14868,0,0,116111,54526,0,0,
%U A144553 438465,201527,0,0,1663720,750169,0,0,6342211,2809931,0,0,24273767
%N A144553 Number of chiral pairs of polyominoes with n cells that have precisely the symmetry group of order 4 generated by 90-degree rotations.
%C A144553 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
%C A144553 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
%C A144553 Polyominoes with precisely 90-degree symmetry centered about square centers and vertices are enumerated by A351142 and A234007 respectively. - _John Mason_, Feb 17 2022
%H A144553 Robert A. Russell, <a href="/A144553/b144553.txt">Table of n, a(n) for n = 1..95</a>
%H A144553 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/animals.html">Enumeration of polyominoes</a>
%H A144553 D. H. Redelmeier, <a href="http://dx.doi.org/10.1016/0012-365X(81)90237-5">Counting polyominoes: yet another attack</a>, Discrete Math., 36 (1981), 191-203.
%H A144553 D. H. Redelmeier, <a href="/A056877/a056877.png">Table 3</a> of Counting polyominoes...
%H A144553 Robert A. Russell, <a href="/A144553/a144553_1.txt">C++ Program</a>
%H A144553 Toshihiro Shirakawa, <a href="https://www.gathering4gardner.org/g4g10gift/math/Shirakawa_Toshihiro-Harmonic_Magic_Square.pdf">Enumeration of Polyominoes considering the symmetry</a>, April 2012, pp. 3-4.
%F A144553 a(n) = A030228(n) - A006747(n) - A006749(n). - _Jean-François Alcover_, Sep 09 2019, after _Andrew Howroyd_ in A030228.
%F A144553 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
%F A144553 a(n) = A351142(n) + A234007(n/4) if n is a multiple of 4, otherwise a(n) = A351142(n). - _John Mason_, Feb 17 2022
%e A144553 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
%t A144553 A006747 = Cases[Import["https://oeis.org/A006747/b006747.txt",
%t A144553      "Table"], {_, _}][[All, 2]];
%t A144553 A006749 = Cases[Import["https://oeis.org/A006749/b006749.txt",
%t A144553      "Table"], {_, _}][[All, 2]];
%t A144553 A030228 = Cases[Import["https://oeis.org/A030228/b030228.txt",
%t A144553      "Table"], {_, _}][[All, 2]];
%t A144553 a[n_] := A030228[[n+1]] - A006747[[n]] - A006749[[n]];
%t A144553 Array[a, 43] (* _Jean-François Alcover_, Sep 09 2019, updated Aug 17 2022 *)
%Y A144553 Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351142, A234007.
%Y A144553 Cf. A324408, A324409 (inner rings).
%Y A144553 Cf. A348848 (C90), A348849 (F90).
%K A144553 nonn
%O A144553 1,12
%A A144553 _N. J. A. Sloane_, Jan 01 2009
%E A144553 a(28) added by _Andrew Howroyd_, Dec 04 2018
%E A144553 a(29)-a(91) added by _Robert A. Russell_, May 23 2020
%E A144553 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
%E A144553 After _John Mason_'s Apr 15 2023 correction to the b-file of A006749, the discrepancy disappeared. - _Andrey Zabolotskiy_, Jan 18 2024