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

1,3

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.

Links

Table of n, a(n) for n=1..35.

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Robert A. Russell, Mapping fixed polyominoes

Example

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

Crossrefs

Cf. A001207, A006535.

Sequence in context: A214319 A062092 A320172 * A079117 A360881 A030137

Adjacent sequences: A350240 A350241 A350242 * A350244 A350245 A350246

Keyword

nonn

Author

Robert A. Russell, Dec 21 2021

Status

approved