A348404 -id:A348404 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

			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".

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).

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...

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.

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

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

A348402 Number of unoriented polyomino rings of length 2n with twofold rotational symmetry.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 3, 9, 13, 35, 59, 147, 280, 669, 1347, 3142, 6545, 15110, 32057, 73625, 158056, 362280, 783800, 1795134, 3906573, 8946154, 19558340

Offset: 1

John Mason, Oct 18 2021

This sequence and its chiral and achiral versions correspond to Robert A. Russell's similar sequences for rings of fourfold rotational symmetry. The sequence does not count the mononimo or domino, referred to by Redelmeier as degenerate rings, as they are not in fact rings.

The sequence refers to rings with at least twofold (180-degree) rotational symmetry, and so includes those with (i) fourfold (90-degree) rotational symmetry, and (ii) all symmetries. - John Mason, Jan 19 2023

			a(2)=1 because of:
  OO
  OO
a(4)=1 because of:
  OOO
  O.O
  OOO
a(5)=1 because of:
  OOOO
  O..O
  OOOO

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

Cf. A348403 (chiral), A348404 (achiral), A324407 (unoriented with fourfold rotational symmetry), A324408 (chiral with fourfold rotational symmetry), A324409 (achiral with fourfold rotational symmetry).

a(n) = A348403(n) + A348404(n).

A348403 Number of chiral pairs of polyomino rings of length 2n with twofold rotational symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 9, 21, 51, 117, 262, 598, 1307, 2987, 6456, 14752, 31859, 72839, 157611, 360472, 782802, 1791140, 3904323, 8936996, 19553272

Offset: 1

John Mason, Oct 18 2021

			a(7)=1 because of:
  OOOO
  O..OO
  OO..O
  .OOOO

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

Cf. A348402 (all unoriented), A348404 (achiral), A324407 (unoriented with fourfold rotational symmetry), A324408 (chiral with fourfold rotational symmetry), A324409 (achiral with fourfold rotational symmetry).

a(n) = A348402(n) - A348404(n).

cp's OEIS Frontend

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).

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A348402 Number of unoriented polyomino rings of length 2n with twofold rotational symmetry.

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A348403 Number of chiral pairs of polyomino rings of length 2n with twofold rotational symmetry.

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