login
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 an asymmetric tile.
6

%I #17 Jul 09 2024 08:56:34

%S 1,6,6,16,76,16,72,1056,1056,72,256,16576,65536,16576,256,1056,262656,

%T 4196352,4196352,262656,1056,4096,4197376,268435456,1073790976,

%U 268435456,4197376,4096,16512,67117056,17180000256,274878431232,274878431232,17180000256,67117056,16512

%N 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 an asymmetric tile.

%H Peter Kagey, <a href="/A368220/a368220.pdf">Illustration of T(2,2)=76</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

%e ----+-----------------------------------------------------------

%e 1 | 1 6 16 72 256

%e 2 | 6 76 1056 16576 262656

%e 3 | 16 1056 65536 4196352 268435456

%e 4 | 72 16576 4196352 1073790976 274878431232

%e 5 | 256 262656 268435456 274878431232 281474976710656

%e 6 | 1056 4197376 17180000256 70368756760576 288230376688582656

%t A368220[n_, m_] := 2^(n*m - 2)*(2^(n*m) + Boole[EvenQ[n*m]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])

%Y Cf. A117401, A225910, A368218, A368219, A368222, A368224.

%K nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 18 2023