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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.
8

%I #24 Jul 06 2024 10:19:44

%S 1,39,32896,536895552,140737496743936,590295810384475521024,

%T 39614081257132309534260330496,42535295865117307939839354957685850112,

%U 730750818665451459101843020821051317142553624576,200867255532373784442745261543120694290360960529885344825344

%N 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.

%C 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.

%H Peter Kagey, <a href="/A367524/a367524.pdf">Illustration of a(2)=39</a>

%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv: 2311.13072 [math.CO], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kagey/kagey6.html">J. Int. Seq.</a>, (2024) Vol. 27, Art. No. 24.6.1, pp. A-6, A-7, A-8.

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

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

%t 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

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

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 10 2023