A367524 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under horizontal reflection, but no other symmetries of the square.

1, 39, 32896, 536895552, 140737496743936, 590295810384475521024, 39614081257132309534260330496, 42535295865117307939839354957685850112, 730750818665451459101843020821051317142553624576, 200867255532373784442745261543120694290360960529885344825344

Offset: 1

Author

Peter Kagey, Dec 10 2023

Keywords

• nonn

Comments

Also, this is the number ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under 180-degree rotation, but no other symmetries of the square.

Links

• Peter Kagey, Illustration of a(2)=39
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-6, A-7, A-8.

Crossrefs

Cf. A054247, A295229, A302484, A367522, A367523, A367525.

Programs

Mathematica

Table[{2^(4 m^2 - 4 m - 2) (2 + 2^(2 m - 1)^2), 2^(2 m^2 - 3) (2 + 3*4^m^2 + 64^m^2)}, {m, 1, 5}] // Flatten

Formula

a(2m-1) = 2^(4m^2 - 4m - 2)*(2 + 2^(2m-1)^2).

a(2m) = 2^(2m^2 - 3)*(2 + 3*4^m^2 + 64^m^2).