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%I #20 Apr 21 2020 03:06:57
%S 3,3,1,6,3,13,9,30,29,78,93,224,315,687,1095,2250,3855,7685,13797,
%T 27012,49939,96909,182361,352698,671091,1296858,2485533,4806078,
%U 9256395,17920860,34636833,67159050,130150587,252745368
%N Number of n-bead necklaces of 3 colors allowing reversal, with no adjacent beads having the same color.
%H Andrew Howroyd, <a href="/A208539/b208539.txt">Table of n, a(n) for n = 1..100</a>
%H Marko Riedel et al., math.stackexchange, <a href="https://math.stackexchange.com/questions/2164399/">Proper colorings of necklaces</a>
%H Marko Riedel et al., math.stackexchange, <a href="https://math.stackexchange.com/questions/3633494/">Proper colorings of bracelets</a>
%F a(2*n+1) = A106365(2*n+1)/2 for n > 0, a(2*n) = (A106365(2*n) + 3*2^(n-1))/2. - _Andrew Howroyd_, Mar 12 2017
%e All solutions for n=4:
%e ..1....1....1....1....1....2
%e ..3....2....3....2....2....3
%e ..2....3....1....1....1....2
%e ..3....2....3....2....3....3
%t T[n_, k_] := If[n == 1, k, (DivisorSum[n, EulerPhi[n/#]*(k - 1)^# &]/n + If[OddQ[n], 1 - k, k*(k - 1)^(n/2)/2])/2]; a[n_] = T[n, 3]; Array[a, 34] (* _Jean-François Alcover_, Nov 01 2017, after _Andrew Howroyd_ *)
%Y Column 3 of A208544.
%K nonn
%O 1,1
%A _R. H. Hardin_, Feb 27 2012