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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 reflection by an asymmetric tile.
3

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

%S 1,2,2,4,6,2,8,20,12,4,16,72,88,39,4,32,272,688,538,104,9,64,1056,

%T 5472,8292,3280,366,10,128,4160,43712,131464,104864,22028,1172,22,256,

%U 16512,349568,2098704,3355456,1399512,149800,4179,30

%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflection by an asymmetric tile.

%H Peter Kagey, <a href="/A368259/a368259.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 2 4 8 16 32

%e 2 | 2 6 20 72 272 1056

%e 3 | 2 12 88 688 5472 43712

%e 4 | 4 39 538 8292 131464 2098704

%e 5 | 4 104 3280 104864 3355456 107374208

%e 6 | 9 366 22028 1399512 89489584 5726711136

%t A368259[n_,m_]:=1/(2n) (DivisorSum[n,EulerPhi[#]*2^(n*m/#)&]+n*2^(n*m/2-1)*Boole[EvenQ[n]])

%K nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 21 2023