A208540 Number of n-bead necklaces of 4 colors allowing reversal, with no adjacent beads having the same color.

4, 6, 4, 21, 24, 92, 156, 498, 1096, 3210, 8052, 22913, 61320, 173088, 478316, 1351983, 3798240, 10781954, 30585828, 87230157, 249056136, 713387076, 2046590844, 5884491500, 16945772208, 48883660146, 141214768972

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) = A106366(2*n+1)/2 for n > 0, a(2*n) = (A106366(2*n) + 2*3^n)/2. - Andrew Howroyd, Mar 12 2017

Example

All solutions for n=3
..1....1....2....1
..2....2....3....3
..3....4....4....4

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, 4]; Array[a, 27] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

Crossrefs

Column 4 of A208544.

Sequence in context: A185145 A258836 A292387 * A019195 A019089 A173462

Adjacent sequences: A208537 A208538 A208539 * A208541 A208542 A208543

Keyword

nonn

Author

R. H. Hardin, Feb 27 2012

Status

approved