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A368264 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder by two distinct tiles.

%I A368264 #11 Oct 19 2024 21:54:51
%S A368264 2,4,3,8,10,4,16,36,24,6,32,136,176,70,8,64,528,1376,1044,208,14,128,
%T A368264 2080,10944,16456,6560,700,20,256,8256,87424,262416,209728,43800,2344,
%U A368264 36,512,32896,699136,4195360,6710912,2796976,299600,8230,60
%N A368264 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder by two distinct tiles.
%H A368264 Peter Kagey, <a href="/A368264/a368264.pdf">Illustration of T(2,3)=36</a>
%H A368264 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 A368264 Table begins:
%e A368264   n\k|  1   2     3       4         5           6
%e A368264   ---+-------------------------------------------
%e A368264    1 |  2   4     8      16        32          64
%e A368264    2 |  3  10    36     136       528        2080
%e A368264    3 |  4  24   176    1376     10944       87424
%e A368264    4 |  6  70  1044   16456    262416     4195360
%e A368264    5 |  8 208  6560  209728   6710912   214748416
%e A368264    6 | 14 700 43800 2796976 178962784 11453291200
%t A368264 A368264[n_, m_] := 1/n (DivisorSum[n, EulerPhi[#]*2^(n*m/#) &])
%Y A368264 Cf. A117401, A368257, A368259, A368261, A368263.
%K A368264 nonn,tabl
%O A368264 1,1
%A A368264 _Peter Kagey_, Dec 21 2023