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A368219 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflection.

%I A368219 #19 Jul 09 2024 09:00:57
%S A368219 1,2,2,3,7,3,6,20,20,6,10,76,136,76,10,20,272,1056,1056,272,20,36,
%T A368219 1072,8256,16576,8256,1072,36,72,4160,65792,262656,262656,65792,4160,
%U A368219 72,136,16576,524800,4197376,8390656,4197376,524800,16576,136
%N A368219 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflection.
%H A368219 Peter Kagey, <a href="/A368219/a368219.pdf">Illustration of T(3,2)=20</a>
%H A368219 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-1, A-3.
%e A368219 Table begins:
%e A368219   n\k |  1    2     3       4         5           6
%e A368219   ----+--------------------------------------------
%e A368219     1 |  1    2     3       6        10          20
%e A368219     2 |  2    7    20      76       272        1072
%e A368219     3 |  3   20   136    1056      8256       65792
%e A368219     4 |  6   76  1056   16576    262656     4197376
%e A368219     5 | 10  272  8256  262656   8390656   268451840
%e A368219     6 | 20 1072 65792 4197376 268451840 17180065792
%t A368219 A368219[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + If[EvenQ[n*m], 1, Sqrt[2]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])
%Y A368219 Cf. A225910, A368218, A368220, A368223.
%K A368219 nonn,tabl
%O A368219 1,2
%A A368219 _Peter Kagey_, Dec 18 2023