A106367 Necklaces with n beads of 5 colors, no 2 adjacent beads the same color.

5, 10, 20, 70, 204, 700, 2340, 8230, 29140, 104968, 381300, 1398500, 5162220, 19175140, 71582940, 268439590, 1010580540, 3817763740, 14467258260, 54975633976, 209430787820, 799645010860, 3059510616420, 11728124734500

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 (5, 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) * (4*(-1)^d + 4^d) for n > 1. - Andrew Howroyd, Mar 12 2017

Mathematica

a[n_] := If[n==1, 5, Sum[EulerPhi[n/d]*(4*(-1)^d+4^d), {d, Divisors[n]}]/n ];
Array[a, 35] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)

Prog

(PARI) a(n) = if(n==1, 5, sumdiv(n, d, eulerphi(n/d)*(4*(-1)^d + 4^d))/n); \\ Andrew Howroyd, Oct 14 2017

Crossrefs

Column 5 of A208535.

Cf. A000031, A001869.

Sequence in context: A229171 A092407 A056496 * A244026 A244022 A365112

Adjacent sequences: A106364 A106365 A106366 * A106368 A106369 A106370

Keyword

nonn

Author

Christian G. Bower, Apr 29 2005

Status

approved