A030223 - OEIS

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

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OFFSET

1,4

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

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)

CROSSREFS

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

Calculation components: A364485, A364486, A364487.

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