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A368305 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.

%I A368305 #12 Oct 19 2024 21:54:51
%S A368305 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 A368305 2386,1184,237,18,36,1182,7700,27936,26892,7700,687,30,60,4150,51112,
%U A368305 361244,674384,354680,50628,2299,46
%N A368305 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 A368305 Peter Kagey, <a href="/A368305/a368305.pdf">Illustration of T(3,3)=44</a>
%H A368305 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 A368305 Table begins:
%e A368305   n\k|  1   2    3      4        5         6
%e A368305   ---+--------------------------------------
%e A368305    1 |  2   3    4      6        8        14
%e A368305    2 |  3   7   14     40      108       362
%e A368305    3 |  4  13   44    218     1200      7700
%e A368305    4 |  6  34  226   2386    27936    361244
%e A368305    5 |  8  78 1184  26892   674384  17920876
%e A368305    6 | 13 237 7700 354680 17950356 955180432
%t A368305 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 A368305 Cf. A368221, A368258, A368302, A368306.
%K A368305 nonn,tabl
%O A368305 1,1
%A A368305 _Peter Kagey_, Dec 21 2023