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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 vertical reflections by two tiles that are each fixed under vertical reflection.
2

%I #9 Oct 19 2024 21:54:51

%S 2,3,3,6,7,4,10,24,14,6,20,76,100,40,8,36,288,700,564,108,14,72,1072,

%T 5560,8296,3384,362,20,136,4224,43800,131856,104968,22288,1182,36,272,

%U 16576,350256,2098720,3358736,1399176,150972,4150,60

%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflections by two tiles that are each fixed under vertical reflection.

%H Peter Kagey, <a href="/A368260/a368260.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 14 100 700 5560 43800

%e 4 | 6 40 564 8296 131856 2098720

%e 5 | 8 108 3384 104968 3358736 107377488

%e 6 | 14 362 22288 1399176 89505984 5726689312

%t A368260[n_, m_] := 1/(2 n) (DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*2^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*2^((n*m - n)/LCM[#, 2])*2^(n/#) &]])

%K nonn,tabl

%O 1,1

%A _Peter Kagey_, Dec 21 2023