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The number of ways of tiling the n X n grid up to 90-degree rotation by a tile that is not fixed under 180-degree rotation.
3

%I #20 Jul 08 2024 10:38:16

%S 1,70,65536,1073758336,281474976710656,1180591620734591303680,

%T 79228162514264337593543950336,85070591730234615870455337878516924416,

%U 1461501637330902918203684832716283019655932542976,401734511064747568885490523085607563280607806359022338048000

%N The number of ways of tiling the n X n grid up to 90-degree rotation by a tile that is not fixed under 180-degree rotation.

%H Peter Kagey, <a href="/A367532/a367532.pdf">Illustration of a(2)=70</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-10.

%F a(2*n-1) = 256^(n^2 - n).

%F a(2*n) = 4^(n^2 - 1)*(2 + 4^n^2 + 64^n^2).

%t Table[{256^(m^2 - m), 4^(m^2 - 1)*(2 + 4^m^2 + 64^m^2)}, {m, 1, 5}] // Flatten

%Y Cf. A047937, A367525, A367529, A367531.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 11 2023