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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 two tiles that are fixed under these reflections.
3

%I #18 Oct 19 2024 21:54:51

%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 downward antidiagonals: 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