|
%I #12 Dec 21 2023 21:12:59
%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 antidiagonals downward: 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
|
