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Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by an asymmetric tile.
5

%I #14 Jan 04 2024 21:13:38

%S 1,4,4,6,28,6,23,194,194,23,52,2196,7296,2196,52,194,26524,350573,

%T 350573,26524,194,586,351588,17895736,67136624,17895736,351588,586,

%U 2131,4798174,954495904,13744131446,13744131446,954495904,4798174,2131

%N Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by an asymmetric tile.

%H Peter Kagey, <a href="/A368304/a368304.pdf">Illustration of T(2,2)=28</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

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

%e 1 | 1 4 6 23 52

%e 2 | 4 28 194 2196 26524

%e 3 | 6 194 7296 350573 17895736

%e 4 | 23 2196 350573 67136624 13744131446

%e 5 | 52 26524 17895736 13744131446 11258999068672

%e 6 | 194 351588 954495904 2932037300956 9607679419823148

%t A368304[n_,m_]:=1/(4*n*m) (DivisorSum[n, Function[d,DivisorSum[m,Function[c,EulerPhi[c]EulerPhi[d]4^(m*n/LCM[c,d])]]]]+If[EvenQ[n],n/2*DivisorSum[m, EulerPhi[#](4^(n*m/LCM[2,#])+4^((n-2)*m/LCM[2,#])*4^(2m/#)*Boole[EvenQ[#]])&],n*DivisorSum[m,EulerPhi[#](4^(n*m/#))&,EvenQ]]+If[EvenQ[m], m/2*DivisorSum[n,EulerPhi[#](4^(n*m/LCM[2,#])+4^((m-2)*n/LCM[2,#])*4^(2n/#)*Boole[EvenQ[#]])&],m*DivisorSum[n, EulerPhi[#](4^(m*n/#))&,EvenQ]]+Which[EvenQ[n]&&EvenQ[m],(n*m)/4 (3*2^(n*m)),OddQ[n*m],0,OddQ[n+m],(n*m)/2 (2^(n*m))])

%Y Cf. A222188, A368220, A368257, A368302, A368303, A368306, A368308, A184271.

%K nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 21 2023