cp's OEIS Frontend

A368306 Table read by downward antidiagonals: 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.

%I A368306 #13 Oct 19 2024 21:54:51
%S A368306 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 A368306 1096,205,10,20,596,7356,26296,26380,7356,623,22,30,2102,49940,350316,
%U A368306 671104,350584,49940,2171,30
%N A368306 Table read by downward antidiagonals: 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 A368306 Peter Kagey, <a href="/A368306/a368306.pdf">Illustration of T(3,3)=32</a>
%H A368306 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 A368306 Table begins:
%e A368306   n\k| 1   2    3      4        5         6
%e A368306   ---+-------------------------------------
%e A368306    1 | 1   2    2      4        4         8
%e A368306    2 | 2   5    8     24       56       190
%e A368306    3 | 2   9   32    186     1096      7356
%e A368306    4 | 4  26  182   2130    26296    350316
%e A368306    5 | 4  62 1096  26380   671104  17899020
%e A368306    6 | 9 205 7356 350584 17897924 954481360
%t A368306 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 A368306 Cf. A368222, A368259, A368304, A368305, A368308, A184271.
%K A368306 nonn,tabl
%O A368306 1,2
%A A368306 _Peter Kagey_, Dec 21 2023