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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.
4

%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