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A368308 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.

%I A368308 #12 Dec 21 2023 21:13:15
%S A368308 1,2,2,2,5,2,4,9,9,4,4,26,32,26,4,9,62,192,192,62,9,10,205,1096,2174,
%T A368308 1096,205,10,22,623,7440,26500,26500,7440,623,22,30,2171,49940,351336,
%U A368308 671104,351336,49940,2171,30
%N A368308 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.
%H A368308 Peter Kagey, <a href="/A368308/a368308.pdf">Illustration of T(3,3)=32</a>
%H A368308 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 A368308 Table begins:
%e A368308   n\k| 1   2    3      4        5         6
%e A368308   ---+-------------------------------------
%e A368308    1 | 1   2    2      4        4         9
%e A368308    2 | 2   5    9     26       62       205
%e A368308    3 | 2   9   32    192     1096      7440
%e A368308    4 | 4  26  192   2174    26500    351336
%e A368308    5 | 4  62 1096  26500   671104  17904476
%e A368308    6 | 9 205 7440 351336 17904476 954546880
%t A368308 A368308[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], 0, OddQ[n + m], 1/2, True, 3/4])
%Y A368308 Cf. A184271, A368224, A368263, A368304, A368306, A368307.
%K A368308 nonn,tabl
%O A368308 1,2
%A A368308 _Peter Kagey_, Dec 21 2023