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%I #64 Apr 15 2023 12:27:20
%S 1,1,1,2,3,6,10,20,34,70,121,250,441,912,1630,3375,6092,12624,22961,
%T 47616,87136,180811,332549,690398,1275166,2648422,4909364,10199792,
%U 18966700,39416488,73497642,152777230,285569898,593717419,1112188817,2312672439,4340728280
%N Number of achiral polyominoes with n cells.
%C Polyominoes with n cells and at least one line of reflection symmetry. - _Joshua Zucker_, Mar 08 2008
%C This sequence can most readily be calculated by enumerating fixed polyominoes for three different axes of symmetry: 1) a line composed of the diagonals of cells, A346800, 2) a line composed of edges of cells, and 3) a line composed of lines connecting the centers of adjacent cells, A346799. For the second case, any fixed polyomino just touching the edge line is reflected on the other side, so that sequence is A001168(n/2) for even values of n and zero otherwise. These three sequences together include each achiral polyomino exactly twice. - _Robert A. Russell_, Aug 04 2021
%H John Mason, <a href="/A030227/b030227.txt">Table of n, a(n) for n = 0..50</a> (terms 1..48 from Robert A. Russell.)
%H 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 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 a(n) = A000105(n) - A030228(n) = 2*A000105(n) - A000988(n). - _Andrew Howroyd_, Dec 04 2018
%F a(n) = A006746(n) + A006748(n) + A056877(n) + A056878(n) + A142886(n) = A000988(n) - 2*A030228(n). - _Robert A. Russell_, Feb 02 2019
%F For odd n, a(n) = (A346799(n) + A346800(n)) / 2; for even n, a(n) = (A346799(n) + A001168(n/2) + A346800(n)) / 2. - _Robert A. Russell_, Aug 04 2021
%e For a(4)=3, the achiral tetrominoes are a 2 X 2 square, a 1 X 4 rectangle, and a cell plus three cells adjacent to it (forming a shortened T).
%t A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {_, _}][[All, 2]];
%t A000988 = Cases[Import["https://oeis.org/A000988/b000988.txt", "Table"], {_, _}][[All, 2]];
%t a[n_] := 2*A000105[[n + 1]] - A000988[[n]];
%t Array[a, 45] (* _Jean-François Alcover_, Sep 08 2019, after _Andrew Howroyd_ *)
%Y Cf. A000988 (oriented), A000105 (unoriented), A030228 (chiral).
%Y Cf. A006746, A006748, A056877, A056878, A142886 (subcategories).
%Y Cf. A001168, A346799, A346800.
%K nonn
%O 0,4
%A _David W. Wilson_
%E a(23)-a(36) from _Andrew Howroyd_, Dec 04 2018
%E Name edited by _Robert A. Russell_, Feb 03 2019
%E Offset changed to 0, and a(0) added by _John Mason_, Jan 12 2023