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A368253 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 two tiles that are fixed under these reflections.

%I A368253 #18 Oct 19 2024 21:54:51
%S A368253 2,3,3,6,7,4,10,24,13,6,20,76,74,34,8,36,288,430,378,78,13,72,1072,
%T A368253 3100,4756,1884,237,18,136,4224,23052,70536,53764,11912,687,30,272,
%U A368253 16576,179736,1083664,1689608,709316,77022,2299,46
%N A368253 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 two tiles that are fixed under these reflections.
%H A368253 Peter Kagey, <a href="/A368253/a368253.pdf">Illustration of T(2,3)=24</a>
%H A368253 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 A368253 Table begins:
%e A368253   n\k |  1   2     3      4        5          6
%e A368253   ----+----------------------------------------
%e A368253     1 |  2   3     6     10       20         36
%e A368253     2 |  3   7    24     76      288       1072
%e A368253     3 |  4  13    74    430     3100      23052
%e A368253     4 |  6  34   378   4756    70536    1083664
%e A368253     5 |  8  78  1884  53764  1689608   53762472
%e A368253     6 | 13 237 11912 709316 44900448 2865540112
%t A368253 A368253[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*If[EvenQ[n], 1/2 (2^((n*m + 2 m)/2) + 2^(n*m/2)), 2^((n*m + 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 - n)/LCM[d, 2])*2^(n/d)]]] + n*Which[EvenQ[m], 2^(n*m/2), OddQ[m] && EvenQ[n], (3/2*2^(n*m/2)), OddQ[m] && OddQ[n], 2^((n*m + 1)/2)])
%Y A368253 Cf. A222188, A225910.
%Y A368253 Cf. A005418 (n=1), A225826 (n=2), A000029 (k=1), A222187 (k=2).
%K A368253 nonn,tabl
%O A368253 1,1
%A A368253 _Peter Kagey_, Dec 19 2023