login
Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.
4

%I #12 Dec 21 2023 21:13:15

%S 1,2,2,2,5,2,4,9,9,4,4,26,32,26,4,9,62,192,192,62,9,10,205,1096,2174,

%T 1096,205,10,22,623,7440,26500,26500,7440,623,22,30,2171,49940,351336,

%U 671104,351336,49940,2171,30

%N Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.

%H Peter Kagey, <a href="/A368308/a368308.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 9

%e 2 | 2 5 9 26 62 205

%e 3 | 2 9 32 192 1096 7440

%e 4 | 4 26 192 2174 26500 351336

%e 5 | 4 62 1096 26500 671104 17904476

%e 6 | 9 205 7440 351336 17904476 954546880

%t A368308[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*m*2^(n*m/2)*Which[OddQ[n*m], 0, OddQ[n + m], 1/2, True, 3/4])

%Y Cf. A184271, A368224, A368263, A368304, A368306, A368307.

%K nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 21 2023