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

Author

N. J. A. Sloane, Jan 01 2009

Keywords

• nonn

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

Examples

			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.
		

Links

• Robert A. Russell, Table of n, a(n) for n = 0..163
• 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...
• Index entries for sequences related to groups

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).

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

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