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%I #21 Aug 06 2023 11:55:33
%S 1,1,1,2,2,5,5,12,13,30,36,80,97,213,266,578,737,1589,2051,4408,5747,
%T 12333,16213,34737,45979,98367,131007,279902,374781,799732,1075793,
%U 2293193,3097415,6596787,8942350,19031088,25880367,55043561,75068945,159570624,218189681
%N Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).
%C These are the achiral polyominoes of the regular tiling with Schläfli symbol {3,6}. An achiral polyomino is identical to its reflection. This sequence can most readily be calculated by enumerating achiral fixed polyominoes for three situations with a given axis of symmetry: 1) fixed polyominoes with an axis of symmetry composed of cell edges, A364485; 2) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and a vertex as the highest polyomino point on this axis, A364486; and 3) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and an edge center as the highest polyomino point on this axis, A364487. Those three sequences include each achiral polyomino exactly twice. - _Robert A. Russell_, Jul 26 2023
%H Robert A. Russell, <a href="/A030223/b030223.txt">Table of n, a(n) for n = 1..60</a>
%H Robert A. Russell, <a href="/A030223/a030223.pdf">Examples for polyominoes with five or fewer cells</a>
%F From _Robert A. Russell_, Jul 27 2023: (Start)
%F a(n) = (A364486(n) + A364487(n)) / 2, n odd.
%F a(n) = (A364485(n/2) + A364486(n) + A364487(n)) / 2, n even.
%F a(n) = 2*A000577(n) - A006534(n) = A006534(n) - 2*A030224(n) = A000577(n) - A030224(n). (End)
%Y Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A001420 (fixed).
%Y Calculation components: A364485, A364486, A364487.
%Y Other tilings: A030227 {4,4}, A030225 {6,3}.
%K nonn
%O 1,4
%A _David W. Wilson_
%E a(19) to a(28) from _Joseph Myers_, Sep 24 2002
%E Additional terms from _Robert A. Russell_, Jul 26 2023
%E Name edited by _Robert A. Russell_, Jul 27 2023
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