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). (Formerly M3741)

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

1,6

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.

Polyominoes having this symmetry may have an axis of symmetry that coincides with 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

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...

Formula

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

Example

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".

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.

Sequence in context: A344435 A100791 A056883 * A184297 A108412 A205008

Adjacent sequences: A006744 A006745 A006746 * A006748 A006749 A006750

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved