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A368303 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 180-degree rotations but not horizontal or vertical reflections.

%I A368303 #12 Oct 19 2024 21:54:51
%S A368303 1,2,2,2,5,2,4,8,8,4,4,22,24,22,4,8,44,120,120,44,8,9,135,612,1203,
%T A368303 612,135,9,18,362,3892,13600,13600,3892,362,18,23,1211,25482,177342,
%U A368303 337600,177342,25482,1211,23,44,3914,176654,2404372,8962618,8962618,2404372,176654,3914,44
%N A368303 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 180-degree rotations but not horizontal or vertical reflections.
%H A368303 Peter Kagey, <a href="/A368303/a368303.pdf">Illustration of T(3,3)=24</a>
%H A368303 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 A368303 Table begins:
%e A368303   n\k| 1   2    3      4       5         6
%e A368303   ---+------------------------------------
%e A368303    1 | 1   2    2      4       4         8
%e A368303    2 | 2   5    8     22      44       135
%e A368303    3 | 2   8   24    120     612      3892
%e A368303    4 | 4  22  120   1203   13600    177342
%e A368303    5 | 4  44  612  13600  337600   8962618
%e A368303    6 | 8 135 3892 177342 8962618 477371760
%t A368303 A368303[n_, m_]:=1/(4*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, Function[c, EulerPhi[c]EulerPhi[d]2^(m*n/LCM[c, d])]]]] + If[EvenQ[n], n/2*DivisorSum[m, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2m/#)*Boole[EvenQ[#]])&], n*DivisorSum[m, EulerPhi[#](2^(n*m/#))&, EvenQ]] + If[EvenQ[m], m/2*DivisorSum[n, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((m - 2)*n/LCM[2, #])*2^(2n/#)*Boole[EvenQ[#]])&], m*DivisorSum[n, EulerPhi[#](2^(m*n/#))&, EvenQ]] + n*m*2^((n*m)/2)*Which[OddQ[n*m], Sqrt[2], OddQ[n + m], 3/2, True, 7/4])
%Y A368303 Cf. A222188, A368219, A368256, A368302, A368304, A368307.
%K A368303 nonn,tabl
%O A368303 1,2
%A A368303 _Peter Kagey_, Dec 21 2023