A364485 Number of fixed triangular polyominoes with 2n cells of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell edges.

1, 2, 4, 9, 23, 59, 155, 418, 1136, 3122, 8663, 24201, 68059, 192471, 546899, 1560511, 4469000, 12839642, 36995629, 106875531, 309477998, 898075778, 2611239508, 7606064348, 22191694916, 64845964156, 189752911736, 555985221037, 1631053277370, 4790356866561

Offset: 1

Author

Robert A. Russell, Jul 26 2023

Keywords

• nonn

Comments

Half of the cells lie on each side of the axis of symmetry, so there must be an even number. This is one of three sequences used to calculate A030223, the number of achiral polyominoes for this tiling. Two fixed polyominoes are identical only if one is a translation of the other.

Examples

			These are halves above the axis of symmetry of the polyominoes for n<4. The bottom edge of each is the axis of symmetry.
                                             /\    /\
          ____  ____   ________   ____      /__\  /__\
   /\    /\  /  \  /\  \  /\  /  /\  /\    /\  /  \  /\
  /__\  /__\/    \/__\  \/__\/  /__\/__\  /__\/    \/__\
		

Crossrefs

Cf. A030223, A364486, A364487.

Formula

a(n) = 2*A030223(2n) - A364486(2n) - A364487(2n).