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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 two tiles that are both fixed under horizontal reflection.
3

%I #12 Oct 19 2024 21:54:51

%S 2,3,3,4,7,4,6,14,13,6,8,40,44,34,8,14,108,218,226,78,13,20,362,1200,

%T 2386,1184,237,18,36,1182,7700,27936,26892,7700,687,30,60,4150,51112,

%U 361244,674384,354680,50628,2299,46

%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 two tiles that are both fixed under horizontal reflection.

%H Peter Kagey, <a href="/A368305/a368305.pdf">Illustration of T(3,3)=44</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 | 2 3 4 6 8 14

%e 2 | 3 7 14 40 108 362

%e 3 | 4 13 44 218 1200 7700

%e 4 | 6 34 226 2386 27936 361244

%e 5 | 8 78 1184 26892 674384 17920876

%e 6 | 13 237 7700 354680 17950356 955180432

%t A368305[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/#))&]/2, DivisorSum[m, EulerPhi[#](2^((n - 1)*m/LCM[2, #])*2^(m/#))&]])

%Y Cf. A368221, A368258, A368302, A368306.

%K nonn,tabl

%O 1,1

%A _Peter Kagey_, Dec 21 2023