A368304 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.
1, 4, 4, 6, 28, 6, 23, 194, 194, 23, 52, 2196, 7296, 2196, 52, 194, 26524, 350573, 350573, 26524, 194, 586, 351588, 17895736, 67136624, 17895736, 351588, 586, 2131, 4798174, 954495904, 13744131446, 13744131446, 954495904, 4798174, 2131
Offset: 1
Examples
Table begins: n\k| 1 2 3 4 5 ---+---------------------------------------------------- 1 | 1 4 6 23 52 2 | 4 28 194 2196 26524 3 | 6 194 7296 350573 17895736 4 | 23 2196 350573 67136624 13744131446 5 | 52 26524 17895736 13744131446 11258999068672 6 | 194 351588 954495904 2932037300956 9607679419823148
Links
- Peter Kagey, Illustration of T(2,2)=28
- Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.
Programs
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Mathematica
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))])