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A208539
Number of n-bead necklaces of 3 colors allowing reversal, with no adjacent beads having the same color.
3
3, 3, 1, 6, 3, 13, 9, 30, 29, 78, 93, 224, 315, 687, 1095, 2250, 3855, 7685, 13797, 27012, 49939, 96909, 182361, 352698, 671091, 1296858, 2485533, 4806078, 9256395, 17920860, 34636833, 67159050, 130150587, 252745368
OFFSET
1,1
LINKS
Marko Riedel et al., math.stackexchange, Proper colorings of necklaces
Marko Riedel et al., math.stackexchange, Proper colorings of bracelets
FORMULA
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
EXAMPLE
All solutions for n=4:
..1....1....1....1....1....2
..3....2....3....2....2....3
..2....3....1....1....1....2
..3....2....3....2....3....3
MATHEMATICA
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 *)
CROSSREFS
Column 3 of A208544.
Sequence in context: A074456 A016454 A065227 * A174128 A131070 A295290
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 27 2012
STATUS
approved