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Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder by two distinct tiles.
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%I #11 Oct 19 2024 21:54:51

%S 2,4,3,8,10,4,16,36,24,6,32,136,176,70,8,64,528,1376,1044,208,14,128,

%T 2080,10944,16456,6560,700,20,256,8256,87424,262416,209728,43800,2344,

%U 36,512,32896,699136,4195360,6710912,2796976,299600,8230,60

%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder by two distinct tiles.

%H Peter Kagey, <a href="/A368264/a368264.pdf">Illustration of T(2,3)=36</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 4 8 16 32 64

%e 2 | 3 10 36 136 528 2080

%e 3 | 4 24 176 1376 10944 87424

%e 4 | 6 70 1044 16456 262416 4195360

%e 5 | 8 208 6560 209728 6710912 214748416

%e 6 | 14 700 43800 2796976 178962784 11453291200

%t A368264[n_, m_] := 1/n (DivisorSum[n, EulerPhi[#]*2^(n*m/#) &])

%Y Cf. A117401, A368257, A368259, A368261, A368263.

%K nonn,tabl

%O 1,1

%A _Peter Kagey_, Dec 21 2023