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The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by a tile that is fixed under 180-degree rotations but is not fixed under either reflection.
5

%I #26 Jul 06 2024 10:03:06

%S 1,5,136,16448,8390656,17179934720,140737496743936,

%T 4611686019501129728,604462909807864343166976,

%U 316912650057057631849152512000,664613997892457937028364282443595776,5575186299632655785385110159782807787798528,187072209578355573530071668259090783432992763150336

%N The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by a tile that is fixed under 180-degree rotations but is not fixed under either reflection.

%H Peter Kagey, <a href="/A367528/a367528.pdf">Illustration of a(2)=5</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-9.

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

%F a(2m) = (4^m^2 + 16^m^2)/4.

%t Table[{2^(2 m^2 - 4 m - 1) (4^m + 4^m^2), (4^m^2 + 16^m^2)/4}, {m, 1, 5}] // Flatten

%Y Cf. A367524, A367526, A367527, A367529.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 10 2023