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A368258 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflections by two tiles that are each fixed under horizontal reflection.

%I A368258 #10 Oct 19 2024 21:54:51
%S A368258 2,4,3,8,10,4,16,36,20,6,32,136,120,55,8,64,528,816,666,136,13,128,
%T A368258 2080,5984,9316,3536,430,18,256,8256,45760,139656,106912,23052,1300,
%U A368258 30,512,32896,357760,2164240,3371840,1415896,151848,4435,46
%N A368258 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflections by two tiles that are each fixed under horizontal reflection.
%H A368258 Peter Kagey, <a href="/A368258/a368258.pdf">Illustration of T(2,3)=36</a>
%H A368258 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 A368258 Table begins:
%e A368258   n\k|  1    2      3        4          5            6
%e A368258   ---+------------------------------------------------
%e A368258    1 |  2    4      8       16         32           64
%e A368258    2 |  3   10     36      136        528         2080
%e A368258    3 |  4   20    120      816       5984        45760
%e A368258    4 |  6   55    666     9316     139656      2164240
%e A368258    5 |  8  136   3536   106912    3371840    107505280
%e A368258    6 | 13  430  23052  1415896   89751728   5730905440
%e A368258    7 | 18 1300 151848 19206736 2454791328 314154568000
%t A368258 A368258[n_,m_] := 1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#)&] + n*2^(n*m/2)*If[EvenQ[n], 1/2*(2^m + 1), 2^(m/2)])
%K A368258 nonn,tabl
%O A368258 1,1
%A A368258 _Peter Kagey_, Dec 21 2023