A350243 Number of achiral hexagonal polyominoes with 3n cells and threefold rotational symmetry centered at a vertex.

1, 1, 2, 5, 9, 19, 39, 82, 171, 368, 773, 1678, 3559, 7776, 16601, 36470, 78295, 172720, 372440, 824512, 1784463, 3961869, 8601227, 19143685, 41671452, 92944943, 202787164, 453138925, 990656774, 2217280465, 4856097782, 10884558781, 23876327783, 53585821550, 117713147451

Offset: 1

Author

Robert A. Russell, Dec 21 2021

Keywords

• nonn

Comments

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.

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.

Examples

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

Links

• D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
• Robert A. Russell, Mapping fixed polyominoes

Crossrefs

Cf. A001207, A006535.