A030223 Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).

1, 1, 1, 2, 2, 5, 5, 12, 13, 30, 36, 80, 97, 213, 266, 578, 737, 1589, 2051, 4408, 5747, 12333, 16213, 34737, 45979, 98367, 131007, 279902, 374781, 799732, 1075793, 2293193, 3097415, 6596787, 8942350, 19031088, 25880367, 55043561, 75068945, 159570624, 218189681

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

These are the achiral polyominoes of the regular tiling with Schläfli symbol {3,6}. An achiral polyomino is identical to its reflection. This sequence can most readily be calculated by enumerating achiral fixed polyominoes for three situations with a given axis of symmetry: 1) fixed polyominoes with an axis of symmetry composed of cell edges, A364485; 2) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and a vertex as the highest polyomino point on this axis, A364486; and 3) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and an edge center as the highest polyomino point on this axis, A364487. Those three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Jul 26 2023

Links

• Robert A. Russell, Table of n, a(n) for n = 1..60
• Robert A. Russell, Examples for polyominoes with five or fewer cells

Crossrefs

Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A001420 (fixed).

Calculation components: A364485, A364486, A364487.

Other tilings: A030227 {4,4}, A030225 {6,3}.

Formula

From Robert A. Russell, Jul 27 2023: (Start)

a(n) = (A364486(n) + A364487(n)) / 2, n odd.

a(n) = (A364485(n/2) + A364486(n) + A364487(n)) / 2, n even.

a(n) = 2*A000577(n) - A006534(n) = A006534(n) - 2*A030224(n) = A000577(n) - A030224(n). (End)

Extensions

a(19) to a(28) from Joseph Myers, Sep 24 2002

Additional terms from Robert A. Russell, Jul 26 2023

Name edited by Robert A. Russell, Jul 27 2023