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A368257 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by an asymmetric tile.

%I A368257 #9 Oct 19 2024 21:54:51
%S A368257 1,6,4,16,44,6,72,544,366,23,256,8384,21856,4244,52,1056,131584,
%T A368257 1399512,1050128,52740,194,4096,2100224,89478656,268472384,53687104,
%U A368257 701124,586,16512,33562624,5726711136,68719870208,54975896016,2863399264,9591666,2131
%N A368257 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by an asymmetric tile.
%H A368257 Peter Kagey, <a href="/A368257/a368257.pdf">Illustration of T(2,2)=44</a>
%H A368257 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 A368257 Table begins:
%e A368257   n\k|   1      2          3              4                 5
%e A368257   ---+-------------------------------------------------------
%e A368257    1 |   1      6         16             72               256
%e A368257    2 |   4     44        544           8384            131584
%e A368257    3 |   6    366      21856        1399512          89478656
%e A368257    4 |  23   4244    1050128      268472384       68719870208
%e A368257    5 |  52  52740   53687104    54975896016    56294995342336
%e A368257    6 | 194 701124 2863399264 11728132423744 48038396383286784
%t A368257 A368257[n_, m_] := 1/(4n)*(DivisorSum[n, EulerPhi[#]*4^(n*m/#) &] + n (2^(n*m - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*4^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*4^(n*m/#) &, EvenQ]] + n*2^(n*m)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])
%K A368257 nonn,tabl
%O A368257 1,2
%A A368257 _Peter Kagey_, Dec 21 2023