login
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.
3

%I #19 Jul 08 2024 10:38:13

%S 1,6,136,16456,8390656,17179934976,140737496743936,

%T 4611686019501162496,604462909807864343166976,

%U 316912650057057631849169289216,664613997892457937028364282443595776,5575186299632655785385110159782842147536896,187072209578355573530071668259090783432992763150336

%N 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 Peter Kagey, <a href="/A367531/a367531.pdf">Illustration of a(3)=136</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) = 2^(2n^2 - 4n - 1)*(4^n + 4^n^2).

%F a(2*n) = 2^(n^2 - 2)*(2 + 2^n^2 + 8^n^2).

%t 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 Cf. A047937, A367532.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 11 2023