|
%I #19 Jul 09 2024 09:00:53
%S 1,3,2,4,7,3,10,20,24,6,16,76,144,76,10,36,272,1120,1056,288,20,64,
%T 1072,8448,16576,8320,1072,36,136,4160,66816,262656,263680,65792,4224,
%U 72,256,16576,528384,4197376,8396800,4197376,525312,16576,136
%N Table read by antidiagonals downward: 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 Peter Kagey, <a href="/A368218/a368218.pdf">Illustration of T(3,2)=24</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-1, A-3.
%e Table begins:
%e n\k | 1 2 3 4 5 6
%e ----+--------------------------------------------
%e 1 | 1 3 4 10 16 36
%e 2 | 2 7 20 76 272 1072
%e 3 | 3 24 144 1120 8448 66816
%e 4 | 6 76 1056 16576 262656 4197376
%e 5 | 10 288 8320 263680 8396800 268517376
%e 6 | 20 1072 65792 4197376 268451840 17180065792
%t 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 Cf. A225910, A367524, A368219, A368220, A368221.
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 18 2023
|
