cp's OEIS Frontend

A030227 Number of achiral polyominoes with n cells.

%I A030227 #64 Apr 15 2023 12:27:20
%S A030227 1,1,1,2,3,6,10,20,34,70,121,250,441,912,1630,3375,6092,12624,22961,
%T A030227 47616,87136,180811,332549,690398,1275166,2648422,4909364,10199792,
%U A030227 18966700,39416488,73497642,152777230,285569898,593717419,1112188817,2312672439,4340728280
%N A030227 Number of achiral polyominoes with n cells.
%C A030227 Polyominoes with n cells and at least one line of reflection symmetry. - _Joshua Zucker_, Mar 08 2008
%C A030227 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 A030227 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 A030227 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 A030227 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 A030227 a(n) = A000105(n) - A030228(n) = 2*A000105(n) - A000988(n). - _Andrew Howroyd_, Dec 04 2018
%F A030227 a(n) = A006746(n) + A006748(n) + A056877(n) + A056878(n) + A142886(n) = A000988(n) - 2*A030228(n). - _Robert A. Russell_, Feb 02 2019
%F A030227 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 A030227 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 A030227 A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {_, _}][[All, 2]];
%t A030227 A000988 = Cases[Import["https://oeis.org/A000988/b000988.txt", "Table"], {_, _}][[All, 2]];
%t A030227 a[n_] := 2*A000105[[n + 1]] - A000988[[n]];
%t A030227 Array[a, 45] (* _Jean-François Alcover_, Sep 08 2019, after _Andrew Howroyd_ *)
%Y A030227 Cf. A000988 (oriented), A000105 (unoriented), A030228 (chiral).
%Y A030227 Cf. A006746, A006748, A056877, A056878, A142886 (subcategories).
%Y A030227 Cf. A001168, A346799, A346800.
%K A030227 nonn
%O A030227 0,4
%A A030227 _David W. Wilson_
%E A030227 a(23)-a(36) from _Andrew Howroyd_, Dec 04 2018
%E A030227 Name edited by _Robert A. Russell_, Feb 03 2019
%E A030227 Offset changed to 0, and a(0) added by _John Mason_, Jan 12 2023