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A030225 Number of achiral hexagonal polyominoes with n cells.

%I A030225 #22 Jul 21 2025 14:38:58
%S A030225 1,1,3,4,11,17,46,75,202,341,914,1581,4222,7436,19794,35357,93859,
%T A030225 169558,449039,818793,2163827,3976636,10489341,19406704,51103471,
%U A030225 95099113,250040802,467679257,1227941119,2307128946,6049886572,11412695367,29891913576,56593284153,148067307799
%N A030225 Number of achiral hexagonal polyominoes with n cells.
%C A030225 These are polyominoes of the Euclidean regular tiling of hexagons with Schläfli symbol {6,3}. This sequence can most readily be calculated by enumerating fixed polyominoes for three situations: 1) fixed polyominoes with a horizontal axis of symmetry along an edge of a cell with no cell centered on that axis, A001207(n/2), 2) fixed polyominoes with a horizontal axis of symmetry that is a diagonal of at least one cell, A347258, and 3) fixed polyominoes with a horizontal axis of symmetry that joins the midpoints of opposite edges of at least one cell, A347257. These three sequences include each achiral polyomino exactly twice. - _Robert A. Russell_, Aug 24 2021
%H A030225 Robert A. Russell, <a href="/A030225/b030225.txt">Table of n, a(n) for n = 1..36</a>
%H A030225 Robert A. Russell, <a href="/A030225/a030225.pdf">Examples for polyominoes with four or fewer cells</a>
%F A030225 From _Robert A. Russell_, Aug 24 2021: (Start)
%F A030225 For odd n, a(n) = (A347257(n) + A347258(n)) / 2; for even n, a(n) = (A001207(n/2) + A347257(n) + A347258(n)) / 2.
%F A030225 a(n) = 2*A000228(n) - A006535(n) = A006535(n) - 2*A030226(n) = A000228(n) - A030226(n). (End)
%t A030225 A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
%t A030225 A000228 = A@000228;
%t A030225 A006535 = A@006535;
%t A030225 a[n_] := 2 A000228[[n]] - A006535[[n]];
%t A030225 a /@ Range[20] (* _Jean-François Alcover_, Feb 22 2020 *)
%Y A030225 Cf. A006535 (oriented), A000228 (unoriented), A030226 (chiral).
%Y A030225 Calculation components: A001207, A347257, A347258.
%Y A030225 Other tilings: A030223 {3,6}, A030227 {4,4}.
%K A030225 nonn
%O A030225 1,3
%A A030225 _David W. Wilson_
%E A030225 More terms from _Joseph Myers_, Sep 21 2002
%E A030225 Name edited by _Robert A. Russell_, Aug 24 2021