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Number of achiral hexagonal polyominoes with 3n cells and threefold rotational symmetry centered at a vertex.
1

%I #11 Dec 25 2021 13:51:44

%S 1,1,2,5,9,19,39,82,171,368,773,1678,3559,7776,16601,36470,78295,

%T 172720,372440,824512,1784463,3961869,8601227,19143685,41671452,

%U 92944943,202787164,453138925,990656774,2217280465,4856097782,10884558781,23876327783,53585821550,117713147451

%N Number of achiral hexagonal polyominoes with 3n cells and threefold rotational symmetry centered at a vertex.

%C These are polyominoes of the regular tiling with Schläfli symbol {6,3}. Each has a symmetry group of order 6. This sequence along with five others and A001207 can be used to determine A006535, the number of oriented polyominoes of the {6,3} regular tiling.

%C The sequence is calculated by using Redelmeier's method to generate fixed polyominoes, which are then mapped to one or two of the symmetric polyominoes as shown in the attachment.

%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 Robert A. Russell, <a href="/A350243/a350243.pdf">Mapping fixed polyominoes</a>

%e For a(1)=1, a(2)=1, and a(3)=2, the polyominoes are:

%e X X X X X X

%e X X X X X X X

%e X X X X X X X X X

%e X X X X X

%Y Cf. A001207, A006535.

%K nonn

%O 1,3

%A _Robert A. Russell_, Dec 21 2021