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Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.
4

%I #13 Oct 19 2024 21:54:51

%S 1,2,2,2,5,2,4,8,9,4,4,24,32,26,4,8,56,186,182,62,9,10,190,1096,2130,

%T 1096,205,10,20,596,7356,26296,26380,7356,623,22,30,2102,49940,350316,

%U 671104,350584,49940,2171,30

%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.

%H Peter Kagey, <a href="/A368306/a368306.pdf">Illustration of T(3,3)=32</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 | 1 2 2 4 4 8

%e 2 | 2 5 8 24 56 190

%e 3 | 2 9 32 186 1096 7356

%e 4 | 4 26 182 2130 26296 350316

%e 5 | 4 62 1096 26380 671104 17899020

%e 6 | 9 205 7356 350584 17897924 954481360

%t A368306[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/#)*Boole[EvenQ[#]]) &]/2, DivisorSum[m, EulerPhi[#]*2^(n*m/#) &, EvenQ]])

%Y Cf. A368222, A368259, A368304, A368305, A368308, A184271.

%K nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 21 2023