A054628 Number of n-bead necklaces with 9 colors.

1, 9, 45, 249, 1665, 11817, 88725, 683289, 5381685, 43046889, 348684381, 2852823609, 23535840225, 195528140649, 1634056945605, 13726075481049, 115813764494505, 981010688215689, 8338590871415805, 71097458824894329, 607883273127192897

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Eric Weisstein's World of Mathematics, Necklace.

Index entries for sequences related to necklaces

Formula

a(n) = (1/n)*Sum_{d|n} phi(d)*9^(n/d), n > 0.

G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 9*x^n)/n. - Herbert Kociemba, Nov 02 2016 [corrected by Ilya Gutkovskiy, Apr 17 2021]

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 9^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021

Example

G.f. = 1 + 9*x + 45*x^2 + 249*x^3 + 1665*x^4 + 11817*x^5 + 88725*x^6 + ...

Maple

with(combstruct):A:=[N, {N=Cycle(Union(Z$9))}, unlabeled]: seq(count(A, size=n), n=0..20); # Zerinvary Lajos, Dec 05 2007

Mathematica

mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-9*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
k=9; 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)*9^(n/d))); \\ Altug Alkan, Sep 21 2018

Crossrefs

Column 9 of A075195.

Cf. A054616.

Sequence in context: A270567 A316108 A026092 * A132133 A009410 A290358

Adjacent sequences: A054625 A054626 A054627 * A054629 A054630 A054631

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