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A368259 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.

%I A368259 #9 Oct 19 2024 21:54:51
%S A368259 1,2,2,4,6,2,8,20,12,4,16,72,88,39,4,32,272,688,538,104,9,64,1056,
%T A368259 5472,8292,3280,366,10,128,4160,43712,131464,104864,22028,1172,22,256,
%U A368259 16512,349568,2098704,3355456,1399512,149800,4179,30
%N A368259 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 A368259 Peter Kagey, <a href="/A368259/a368259.pdf">Illustration of T(2,3)=20</a>
%H A368259 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 A368259 Table begins:
%e A368259   n\k| 1   2     3       4        5          6
%e A368259   ---+----------------------------------------
%e A368259    1 | 1   2     4       8       16         32
%e A368259    2 | 2   6    20      72      272       1056
%e A368259    3 | 2  12    88     688     5472      43712
%e A368259    4 | 4  39   538    8292   131464    2098704
%e A368259    5 | 4 104  3280  104864  3355456  107374208
%e A368259    6 | 9 366 22028 1399512 89489584 5726711136
%t A368259 A368259[n_,m_]:=1/(2n) (DivisorSum[n,EulerPhi[#]*2^(n*m/#)&]+n*2^(n*m/2-1)*Boole[EvenQ[n]])
%K A368259 nonn,tabl
%O A368259 1,2
%A A368259 _Peter Kagey_, Dec 21 2023