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

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

Author

John Mason, Oct 18 2021

Keywords

• nonn
• more

Comments

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

Examples

			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
		

Links

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

Crossrefs

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

Formula

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