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

Author

R. H. Hardin, Feb 27 2012

Keywords

• nonn

Examples

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

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

Crossrefs

Column 4 of A208544.

Programs

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 *)

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