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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 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.)