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%I #17 Dec 21 2023 11:03:42
%S 2,3,3,6,7,4,10,24,13,6,20,76,74,34,8,36,288,430,378,78,13,72,1072,
%T 3100,4756,1884,237,18,136,4224,23052,70536,53764,11912,687,30,272,
%U 16576,179736,1083664,1689608,709316,77022,2299,46
%N Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by two tiles that are fixed under these reflections.
%H Peter Kagey, <a href="/A368253/a368253.pdf">Illustration of T(2,3)=24</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 6 10 20 36
%e 2 | 3 7 24 76 288 1072
%e 3 | 4 13 74 430 3100 23052
%e 4 | 6 34 378 4756 70536 1083664
%e 5 | 8 78 1884 53764 1689608 53762472
%e 6 | 13 237 11912 709316 44900448 2865540112
%t A368253[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*If[EvenQ[n], 1/2 (2^((n*m + 2 m)/2) + 2^(n*m/2)), 2^((n*m + m)/2)] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^((n*m - n)/LCM[d, 2])*2^(n/d)]]] + n*Which[EvenQ[m], 2^(n*m/2), OddQ[m] && EvenQ[n], (3/2*2^(n*m/2)), OddQ[m] && OddQ[n], 2^((n*m + 1)/2)])
%Y Cf. A222188, A225910.
%Y Cf. A005418 (n=1), A225826 (n=2), A000029 (k=1), A222187 (k=2).
%K nonn,tabl
%O 1,1
%A _Peter Kagey_, Dec 19 2023
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