A368305 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by two tiles that are both fixed under horizontal reflection.
2, 3, 3, 4, 7, 4, 6, 14, 13, 6, 8, 40, 44, 34, 8, 14, 108, 218, 226, 78, 13, 20, 362, 1200, 2386, 1184, 237, 18, 36, 1182, 7700, 27936, 26892, 7700, 687, 30, 60, 4150, 51112, 361244, 674384, 354680, 50628, 2299, 46
Offset: 1
Examples
Table begins: n\k| 1 2 3 4 5 6 ---+-------------------------------------- 1 | 2 3 4 6 8 14 2 | 3 7 14 40 108 362 3 | 4 13 44 218 1200 7700 4 | 6 34 226 2386 27936 361244 5 | 8 78 1184 26892 674384 17920876 6 | 13 237 7700 354680 17950356 955180432
Links
- Peter Kagey, Illustration of T(3,3)=44
- 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
A368305[n_, m_]:=1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#]EulerPhi[d]2^(m*n/LCM[#, d])&]]] + n*If[EvenQ[n], DivisorSum[m, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*4^(m/#))&]/2, DivisorSum[m, EulerPhi[#](2^((n - 1)*m/LCM[2, #])*2^(m/#))&]])