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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.
3

%I #11 Dec 21 2023 21:13:07

%S 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 2302,1224,237,18,30,687,7696,27012,27012,7696,687,30,46,2299,50964,

%U 353384,675200,353384,50964,2299,46

%N 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 Peter Kagey, <a href="/A368307/a368307.pdf">Illustration of T(3,3)=48</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.

%e Table begins:

%e n\k| 1 2 3 4 5 6

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

%e 1 | 2 3 4 6 8 13

%e 2 | 3 7 13 34 78 237

%e 3 | 4 13 48 224 1224 7696

%e 4 | 6 34 224 2302 27012 353384

%e 5 | 8 78 1224 27012 675200 17920860

%e 6 | 13 237 7696 353384 17920860 954677952

%t 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 Cf. A368223, A368262, A368303, A368308.

%K nonn,tabl

%O 1,1

%A _Peter Kagey_, Dec 21 2023