%I #12 Oct 19 2024 21:54:51
%S 1,2,2,2,5,2,4,9,8,4,4,26,22,22,4,9,62,120,126,44,8,10,205,600,1267,
%T 592,135,9,22,623,3936,14164,13600,3936,362,18,30,2171,25556,181782,
%U 337192,178366,25314,1211,23,62,7429,177678,2437726,8965354,8980642,2404372,176998,3914,44
%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under horizontal reflections but not vertical reflections.
%H Peter Kagey, <a href="/A368302/a368302.pdf">Illustration of T(3,3)=22</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 9
%e 2 | 2 5 9 26 62 205
%e 3 | 2 8 22 120 600 3936
%e 4 | 4 22 126 1267 14164 181782
%e 5 | 4 44 592 13600 337192 8965354
%e 6 | 8 135 3936 178366 8980642 477655760
%t A368302[n_, m_] := 1/(4*n*m) (DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], 1/2*DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2 m/#)) &], DivisorSum[m, EulerPhi[#] (2^((n - 1)*m/LCM[2, #])*2^(m/#)) &]] + m*If[EvenQ[m], 1/2*DivisorSum[n, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^(m*n/#)*Boole[EvenQ[#]]) &], DivisorSum[n, EulerPhi[#]*2^(m*n/#) &, EvenQ]] + n*m*2^(n*m/2)*Which[EvenQ[n] && EvenQ[m], 3/4, OddQ[n*m], 0, OddQ[n + m], 1/2])
%Y Cf. A222188, A368218, A368254, A368303, A368304, A368305.
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 21 2023