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A368218 Table read by downward 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 horizontal reflection only.

%I A368218 #20 Oct 19 2024 21:54:51
%S A368218 1,3,2,4,7,3,10,20,24,6,16,76,144,76,10,36,272,1120,1056,288,20,64,
%T A368218 1072,8448,16576,8320,1072,36,136,4160,66816,262656,263680,65792,4224,
%U A368218 72,256,16576,528384,4197376,8396800,4197376,525312,16576,136
%N A368218 Table read by downward 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 horizontal reflection only.
%H A368218 Peter Kagey, <a href="/A368218/a368218.pdf">Illustration of T(3,2)=24</a>
%H A368218 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 A368218 Table begins:
%e A368218   n\k |  1    2     3       4         5           6
%e A368218   ----+--------------------------------------------
%e A368218     1 |  1    3     4      10        16          36
%e A368218     2 |  2    7    20      76       272        1072
%e A368218     3 |  3   24   144    1120      8448       66816
%e A368218     4 |  6   76  1056   16576    262656     4197376
%e A368218     5 | 10  288  8320  263680   8396800   268517376
%e A368218     6 | 20 1072 65792 4197376 268451840 17180065792
%t A368218 A368218[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + Boole[EvenQ[n*m]] + Boole[EvenQ[m]] + If[EvenQ[n], 1, 2^(m/2)])
%Y A368218 Cf. A225910, A367524, A368219, A368220, A368221.
%K A368218 nonn,tabl
%O A368218 1,2
%A A368218 _Peter Kagey_, Dec 18 2023