A054629 Number of n-bead necklaces with 10 colors.

1, 10, 55, 340, 2530, 20008, 166870, 1428580, 12501280, 111111340, 1000010044, 9090909100, 83333418520, 769230769240, 7142857857190, 66666666680272, 625000006251280, 5882352941176480, 55555555611222370, 526315789473684220

Offset: 0

Author

N. J. A. Sloane, Apr 16 2000

Keywords

• nonn

Examples

			G.f. = 1 + 10*x + 55*x^2 + 340*x^3 + 2530*x^4 + 20008*x^5 + 166870*x^6 + ...
		

Links

• Eric Weisstein's World of Mathematics, Necklace.
• Index entries for sequences related to necklaces

Crossrefs

Column 10 of A075195.

Cf. A054617.

Programs

Maple

with(combstruct):A:=[N,{N=Cycle(Union(Z$10))},unlabeled]: seq(count(A,size=n),n=0..19); # Zerinvary Lajos, Dec 05 2007

Mathematica

mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-10*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
k=10; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* Robert A. Russell, Sep 21 2018 *)

PARI

a(n)=if(n==0, 1, 1/n*sumdiv(n, d, eulerphi(d)*10^(n/d))); \\ Altug Alkan, Sep 21 2018

Formula

a(n) = (1/n)*Sum_{d|n} phi(d)*10^(n/d) = A054617(n)/n, n > 0.

G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 10*x^n)/n. - Herbert Kociemba, Nov 02 2016

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 10^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021

Extensions

Edited by Christian G. Bower, Sep 07 2002

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