A054625 - OEIS

%I #27 Apr 17 2021 09:07:14

%S 1,6,21,76,336,1560,7826,39996,210126,1119796,6047412,32981556,

%T 181402676,1004668776,5597460306,31345666736,176319474366,

%U 995685849696,5642220380006,32071565263716,182807925027504,1044616697187576,5982804736593846

%N Number of n-bead necklaces with 6 colors.

%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Necklace.html">Necklace.</a>

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

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

%F a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 6^gcd(n,k). - _Ilya Gutkovskiy_, Apr 17 2021

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

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

%t 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 *)

%t 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 *)

%Y Column 6 of A075195.

%Y Cf. A054613.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Apr 16 2000

%E Edited by _Christian G. Bower_, Sep 07 2002

%E a(0) corrected by _Herbert Kociemba_, Nov 02 2016