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A368254 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 a tile that is fixed under horizontal reflections but not vertical reflections.

%I A368254 #13 Oct 19 2024 21:54:51
%S A368254 1,3,2,4,7,2,10,20,13,4,16,76,60,34,4,36,272,430,346,78,8,64,1072,
%T A368254 2992,4756,1768,237,9,136,4160,23052,70024,53764,11612,687,18,256,
%U A368254 16576,178880,1083664,1685920,709316,75924,2299,23
%N A368254 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 a tile that is fixed under horizontal reflections but not vertical reflections.
%H A368254 Peter Kagey, <a href="/A368254/a368254.pdf">Illustration of T(2,3)=20</a>
%H A368254 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 A368254 Table begins:
%e A368254   n\k| 1   2     3      4        5          6
%e A368254   ---+---------------------------------------
%e A368254    1 | 1   3     4     10       16         36
%e A368254    2 | 2   7    20     76      272       1072
%e A368254    3 | 2  13    60    430     2992      23052
%e A368254    4 | 4  34   346   4756    70024    1083664
%e A368254    5 | 4  78  1768  53764  1685920   53762472
%e A368254    6 | 8 237 11612 709316 44881328 2865540112
%t A368254 A368254[n_, m_] := 1/(4n)(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*2^(n*m/2)*If[EvenQ[n], 1/2 (2^m + 1), 2^(m/2)] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)], EvenQ]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])
%Y A368254 Cf. A368218, A368253.
%K A368254 nonn,tabl
%O A368254 1,2
%A A368254 _Peter Kagey_, Dec 19 2023