A142886 Number of polyominoes with n cells that have the symmetry group D_8.

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 12, 7, 0, 0, 20, 11, 0, 0, 45, 20, 0, 0, 80, 36, 0, 0, 173, 65, 0, 0, 310, 117, 0, 0, 664, 216, 0, 0, 1210, 396, 0, 0, 2570, 736, 0, 0, 4728, 1369, 0, 0, 9976, 2558, 0, 0, 18468, 4787, 0, 0, 38840

Offset

0,10

Comments

This is the largest possible symmetry group that a polyomino can have.

Polyominoes with such symmetry centered about square centers and vertices are enumerated by A351127 and A346800 respectively. - John Mason, Feb 16 2022

Links

Michael Spiegel, Table of n, a(n) for n = 0..203 (terms 0..163 from Robert A. Russell)

Tomás Oliveira e Silva, Enumeration of polyominoes

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

D. H. Redelmeier, Table 3 of Counting polyominoes...

Michael Spiegel, Rust program

Index entries for sequences related to groups

Formula

a(n) = A351127(n) + A346800(n/4) if n is a multiple of 4, otherwise a(n) = A351127(n). - John Mason, Feb 16 2022

Example

The monomino has eight-fold symmetry. The tetromino with eight-fold symmetry is four cells in a square. The pentomino with eight-fold symmetry is a cell and its four adjacent cells.

Crossrefs

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351127, A346800.

Cf. A376971 (polycubes with full symmetry).

Sequence in context: A225853 A342128 A330463 * A374019 A099026 A341410

Adjacent sequences: A142883 A142884 A142885 * A142887 A142888 A142889

Keyword

nonn

Author

N. J. A. Sloane, Jan 01 2009

Extensions

Name corrected by Wesley Prosser, Sep 06 2017

a(28) added by Andrew Howroyd, Dec 04 2018

More terms from Robert A. Russell, Jan 13 2019

Status

approved