A106366 Necklaces with n beads of 4 colors, no 2 adjacent beads the same color.

4, 6, 8, 24, 48, 130, 312, 834, 2192, 5934, 16104, 44368, 122640, 341802, 956632, 2690844, 7596480, 21524542, 61171656, 174342216, 498112272, 1426419858, 4093181688, 11767920118, 33891544416, 97764131646, 282429537944

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Index entries for sequences related to necklaces

Formula

CycleBG transform of (4, 0, 0, 0, ...)

CycleBG transform T(A) = invMOEBIUS(invEULER(Carlitz(A)) + A(x^2) - A) + A.

Carlitz transform T(A(x)) has g.f. 1/(1-Sum_{k>0} (-1)^(k+1)*A(x^k)).

a(n) = (1/n) * Sum_{d | n} totient(n/d) * (3*(-1)^d + 3^d) for n > 1. - Andrew Howroyd, Mar 12 2017

Mathematica

a[n_] := If[n==1, 4, Sum[EulerPhi[n/d]*(3*(-1)^d+3^d), {d, Divisors[n]}]/n ];
Array[a, 35] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)

Prog

(PARI) a(n) = if(n==1, 4, sumdiv(n, d, eulerphi(n/d)*(3*(-1)^d + 3^d))/n); \\ Andrew Howroyd, Oct 14 2017

Crossrefs

Column 4 of A208535.

Cf. A000031, A001868, A226493.

Sequence in context: A063719 A272862 A361662 * A019160 A126233 A059889

Adjacent sequences: A106363 A106364 A106365 * A106367 A106368 A106369

Keyword

nonn

Author

Christian G. Bower, Apr 29 2005

Status

approved