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A054625 Number of n-bead necklaces with 6 colors. 8
1, 6, 21, 76, 336, 1560, 7826, 39996, 210126, 1119796, 6047412, 32981556, 181402676, 1004668776, 5597460306, 31345666736, 176319474366, 995685849696, 5642220380006, 32071565263716, 182807925027504, 1044616697187576, 5982804736593846 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..22.

Index entries for sequences related to necklaces

Eric Weisstein's World of Mathematics, Necklace.

FORMULA

a(n) = (1/n)*Sum_{d|n} phi(d)*6^(n/d), n > 0.

G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 6*x^n)/n. - Herbert Kociemba, Nov 02 2016

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 6^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021

EXAMPLE

G.f. = 1 + 6*x + 21*x^2 + 76*x^3 + 336*x^4 + 1650*x^5 + 7826*x^6 + 39996*x^7 + ...

MAPLE

with(combstruct):A:=[N, {N=Cycle(Union(Z$6))}, unlabeled]: seq(count(A, size=n), n=0..22); # Zerinvary Lajos, Dec 05 2007

MATHEMATICA

f[n_] := Block[{d = Divisors@ n}, Total[EulerPhi[d]*6^(n/d)]/n]; f[0] = 1; Array[f, 23, 0] (* Robert G. Wilson v, Jan 01 2013 *)

mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-6*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)

CROSSREFS

Column 6 of A075195.

Cf. A054613.

Sequence in context: A027281 A006814 A108136 * A192733 A344205 A192144

Adjacent sequences:  A054622 A054623 A054624 * A054626 A054627 A054628

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Apr 16 2000

EXTENSIONS

Edited by Christian G. Bower, Sep 07 2002

a(0) corrected by Herbert Kociemba, Nov 02 2016

STATUS

approved

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Last modified August 1 01:54 EDT 2021. Contains 346377 sequences. (Running on oeis4.)