%I #9 Oct 19 2024 21:54:51
%S 1,6,4,16,44,6,72,544,366,23,256,8384,21856,4244,52,1056,131584,
%T 1399512,1050128,52740,194,4096,2100224,89478656,268472384,53687104,
%U 701124,586,16512,33562624,5726711136,68719870208,54975896016,2863399264,9591666,2131
%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by an asymmetric tile.
%H Peter Kagey, <a href="/A368257/a368257.pdf">Illustration of T(2,2)=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
%e ---+-------------------------------------------------------
%e 1 | 1 6 16 72 256
%e 2 | 4 44 544 8384 131584
%e 3 | 6 366 21856 1399512 89478656
%e 4 | 23 4244 1050128 268472384 68719870208
%e 5 | 52 52740 53687104 54975896016 56294995342336
%e 6 | 194 701124 2863399264 11728132423744 48038396383286784
%t A368257[n_, m_] := 1/(4n)*(DivisorSum[n, EulerPhi[#]*4^(n*m/#) &] + n (2^(n*m - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*4^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*4^(n*m/#) &, EvenQ]] + n*2^(n*m)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 21 2023