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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
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
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 March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)