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A367531 The number of ways of tiling the n X n grid up to 90-degree rotation by a tile that is fixed under 180-degree rotation but not 90-degree rotation.

%I A367531 #19 Jul 08 2024 10:38:13
%S A367531 1,6,136,16456,8390656,17179934976,140737496743936,
%T A367531 4611686019501162496,604462909807864343166976,
%U A367531 316912650057057631849169289216,664613997892457937028364282443595776,5575186299632655785385110159782842147536896,187072209578355573530071668259090783432992763150336
%N A367531 The number of ways of tiling the n X n grid up to 90-degree rotation by a tile that is fixed under 180-degree rotation but not 90-degree rotation.
%H A367531 Peter Kagey, <a href="/A367531/a367531.pdf">Illustration of a(3)=136</a>
%H A367531 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 A367531 a(2*n-1) = 2^(2n^2 - 4n - 1)*(4^n + 4^n^2).
%F A367531 a(2*n)   = 2^(n^2 - 2)*(2 + 2^n^2 + 8^n^2).
%t A367531 Table[{2^(2 m^2 - 4 m - 1)*(4^m + 4^m^2), 2^(m^2 - 2)*(2 + 2^m^2 + 8^m^2)}, {m, 1, 5}] // Flatten
%Y A367531 Cf. A047937, A367532.
%K A367531 nonn
%O A367531 1,2
%A A367531 _Peter Kagey_, Dec 11 2023