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

0,2

Links

Table of n, a(n) for n=0..19.

Eric Weisstein's World of Mathematics, Necklace.

Index entries for sequences related to necklaces

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

Example

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

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 *)

Prog

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

Crossrefs

Column 10 of A075195.

Cf. A054617.

Sequence in context: A316109 A188168 A199413 * A290359 A030114 A001557

Adjacent sequences: A054626 A054627 A054628 * A054630 A054631 A054632

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