

A030225


Number of achiral hexagonal polyominoes with n cells.


5



1, 1, 3, 4, 11, 17, 46, 75, 202, 341, 914, 1581, 4222, 7436, 19794, 35357, 93859, 169558, 449039, 818793, 2163827, 3976636, 10489341, 19406704, 51103471, 95099113, 250040802, 467679257, 1227941119, 2307128946
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OFFSET

1,3


COMMENTS

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


LINKS

Robert A. Russell, Table of n, a(n) for n = 1..36
Robert A. Russell, Examples for polyominoes with four or fewer cells


FORMULA

From Robert A. Russell, Aug 24 2021: (Start)
For odd n, a(n) = (A347257(n) + A347258(n)) / 2; for even n, a(n) = (A001207(n/2) + A347257(n) + A347258(n)) / 2.
a(n) = 2*A000228(n)  A006535(n) = A006535(n)  2*A030226(n) = A000228(n)  A030226(n). (End)


MATHEMATICA

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
A000228 = A@000228;
A006535 = A@006535;
a[n_] := 2 A000228[[n]]  A006535[[n]];
a /@ Range[20] (* JeanFrançois Alcover, Feb 22 2020 *)


CROSSREFS

Cf. A006535 (oriented), A000228 (unoriented), A030226 (chiral).
Calculation components: A001207, A347257, A347258.
Other tilings: A030223 {3,6}, A030227 {4,4}.
Sequence in context: A026753 A027222 A026380 * A339157 A060285 A025079
Adjacent sequences: A030222 A030223 A030224 * A030226 A030227 A030228


KEYWORD

nonn,more


AUTHOR

David W. Wilson


EXTENSIONS

More terms from Joseph Myers, Sep 21 2002
Name edited by Robert A. Russell, Aug 24 2021


STATUS

approved



