A208539 Number of n-bead necklaces of 3 colors allowing reversal, with no adjacent beads having the same color.

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

Andrew Howroyd, Table of n, a(n) for n = 1..100

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

Adjacent sequences: A208536 A208537 A208538 * A208540 A208541 A208542

Keyword

nonn

Author

R. H. Hardin, Feb 27 2012

Status

approved