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

1,4

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}.

Sequence in context: A002014 A135153 A097896 * A300436 A056504 A122205

Adjacent sequences: A030220 A030221 A030222 * A030224 A030225 A030226

Keyword

nonn

Author

David W. Wilson

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

Status

approved