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A368307 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by two tiles that are both fixed under 180-degree rotation.

%I A368307 #11 Dec 21 2023 21:13:07
%S A368307 2,3,3,4,7,4,6,13,13,6,8,34,48,34,8,13,78,224,224,78,13,18,237,1224,
%T A368307 2302,1224,237,18,30,687,7696,27012,27012,7696,687,30,46,2299,50964,
%U A368307 353384,675200,353384,50964,2299,46
%N A368307 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by two tiles that are both fixed under 180-degree rotation.
%H A368307 Peter Kagey, <a href="/A368307/a368307.pdf">Illustration of T(3,3)=48</a>
%H A368307 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.
%e A368307 Table begins:
%e A368307   n\k|  1   2    3      4        5         6
%e A368307   ---+--------------------------------------
%e A368307    1 |  2   3    4      6        8        13
%e A368307    2 |  3   7   13     34       78       237
%e A368307    3 |  4  13   48    224     1224      7696
%e A368307    4 |  6  34  224   2302    27012    353384
%e A368307    5 |  8  78 1224  27012   675200  17920860
%e A368307    6 | 13 237 7696 353384 17920860 954677952
%t A368307 A368307[n_, m_] :=  1/(2*n*m) (DivisorSum[n,  Function[d,  DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] +  n*m*2^(n*m/2)* Which[OddQ[n*m], Sqrt[2], OddQ[n + m], 3/2, True, 7/4])
%Y A368307 Cf. A368223, A368262, A368303, A368308.
%K A368307 nonn,tabl
%O A368307 1,1
%A A368307 _Peter Kagey_, Dec 21 2023