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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 a tile that is fixed under horizontal reflections but not vertical reflections.
3

%I #13 Oct 19 2024 21:54:51

%S 1,3,2,4,7,2,10,20,13,4,16,76,60,34,4,36,272,430,346,78,8,64,1072,

%T 2992,4756,1768,237,9,136,4160,23052,70024,53764,11612,687,18,256,

%U 16576,178880,1083664,1685920,709316,75924,2299,23

%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 a tile that is fixed under horizontal reflections but not vertical reflections.

%H Peter Kagey, <a href="/A368254/a368254.pdf">Illustration of T(2,3)=20</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 | 1 3 4 10 16 36

%e 2 | 2 7 20 76 272 1072

%e 3 | 2 13 60 430 2992 23052

%e 4 | 4 34 346 4756 70024 1083664

%e 5 | 4 78 1768 53764 1685920 53762472

%e 6 | 8 237 11612 709316 44881328 2865540112

%t A368254[n_, m_] := 1/(4n)(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*2^(n*m/2)*If[EvenQ[n], 1/2 (2^m + 1), 2^(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/d)], EvenQ]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])

%Y Cf. A368218, A368253.

%K nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 19 2023