A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

0, 0, 0, 1, 1, 5, 4, 18, 19, 73, 73, 278, 283, 1076, 1090, 4125, 4183, 15939, 16105, 61628, 62170, 239388, 240907, 932230, 936447, 3641945, 3651618, 14262540, 14277519, 55987858, 55961118, 220223982, 219813564, 867835023, 865091976, 3425442681

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This sequence gives the number of free polyominoes with symmetry group "R" in Redelmeier's notation. See his Tables 1 and 3, also the column "Rot" in Oliveira e Silva's table.

The rotation center of a polyomino with this symmetry may lie at the center of a square, the middle of an edge, or a vertex of a square. These subsets are enumerated by A351615, A234008 and A351616 respectively. - John Mason, Feb 17 2022, reformulated by Günter Rote, Oct 19 2023

Examples

			a(2) = 0 because the "domino" polyomino has symmetry group of order 4.
For n=3, the three-celled polyomino [ | | ] has group of order 4, and the polyomino
. [ ]
. [ | ]
has only reflective symmetry, so a(3) = 0.
a(4) = 1 because of (in Golomb's notation) the "skew tetromino".
		

References

• S. W. Golomb, Polyominoes, Princeton Univ. Press, NJ, 1994.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• John Mason, Table of n, a(n) for n = 1..50
• Tomás Oliveira e Silva, Enumeration of polyominoes
• Tomás Oliveira e Silva, Numbers of polyominoes classified according to Redelmeier's symmetry classes (an extract from the previous link)
• D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
• D. H. Redelmeier, Table 3 of Counting polyominoes...

Crossrefs

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351615, A234008, A351616.

Polyomino rings of length 2n with twofold rotational symmetry: A348402, A348403, A348404.

Formula

a(n) = A351615(n) + A234008(n/2) + A351616(n/2) for even n, otherwise a(n) = A351615(n). - John Mason, Feb 17 2022

Extensions

Extended to n=28 by Tomás Oliveira e Silva

a(1)-a(3) prepended by Andrew Howroyd, Dec 04 2018

Edited by N. J. A. Sloane, Nov 28 2020

a(29)-a(36) from John Mason, Oct 16 2021