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A106368 Necklaces with n beads of 6 colors, no 2 adjacent beads the same color. 4
6, 15, 40, 165, 624, 2635, 11160, 48915, 217040, 976887, 4438920, 20346485, 93900240, 435970995, 2034505656, 9536767665, 44878791360, 211927736135, 1003867701480, 4768372070757, 22706531350480, 108372083629275 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

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

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Vol. 226-228 (1995), 57-72.

Petros Hadjicostas, Proof of an explicit formula for Bower's CycleBG transform.

Index entries for sequences related to necklaces

FORMULA

CycleBG transform of (6, 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)).

General formula for the CycleBG transform: T(A)(x) = A(x) - Sum_{k>=0} A(x^{2k+1}) + Sum_{k>=1} (phi(k)/k)*log(Carlitz(A)(x^k)). For a proof, see the links. (For this sequence, A(x) = 6*x.) - Petros Hadjicostas, Sep 16 2017

a(n) = (1/n) * Sum_{d | n} totient(n/d) * (5*(-1)^d + 5^d) for n > 1. - Andrew Howroyd, Mar 12 2017

G.f.: m*x - Sum_{k>=1} (phi(k)/k)*((m-1)*B(x^k)+C(x^k)) = m*x - (m-1)*x/(1-x^2) - Sum_{k>=1} (phi(k)/k)*C(x^k), where B(x) = log(1+x) and C(x) = log(1-(m-1)*x). Here m=6 is the number of colors. - Petros Hadjicostas, Sep 16 2017

MATHEMATICA

a[1] = 6;

a[n_] := (1/n) Sum[EulerPhi[n/d]*(5*(-1)^d + 5^d), {d, Divisors[n]}];

Array[a, 30] (* Jean-Fran├žois Alcover, Jul 02 2018, after Andrew Howroyd *)

PROG

(PARI) a(n) = if(n==1, 6, sumdiv(n, d, eulerphi(n/d)*(5*(-1)^d + 5^d))/n); \\ Andrew Howroyd, Oct 14 2017

CROSSREFS

Column 6 of A208535.

Cf. A000031.

Sequence in context: A272847 A273829 A114696 * A100491 A220030 A230165

Adjacent sequences:  A106365 A106366 A106367 * A106369 A106370 A106371

KEYWORD

nonn

AUTHOR

Christian G. Bower, Apr 29 2005

STATUS

approved

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Last modified February 16 21:12 EST 2019. Contains 320199 sequences. (Running on oeis4.)