A054626 Number of n-bead necklaces with 7 colors.

1, 7, 28, 119, 616, 3367, 19684, 117655, 720916, 4483815, 28249228, 179756983, 1153450872, 7453000807, 48444564052, 316504102999, 2077058521216, 13684147881607, 90467424361132, 599941851861751, 3989613329006536, 26597422099282535

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Eric Weisstein's World of Mathematics, Necklace.

Index entries for sequences related to necklaces

Formula

a(n) = (1/n)*Sum_{d|n} phi(d)*7^(n/d), n > 0.

G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 7*x^n)/n. - Herbert Kociemba, Nov 02 2016

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 7^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021

Example

G.f. = 1 + 7*x + 28*x^2 + 119*x^3 + 616*x^4 + 3367*x^5 + 19684*x^6 + ...

Maple

with(combstruct):A:=[N, {N=Cycle(Union(Z$7))}, unlabeled]: seq(count(A, size=n), n=0..21); # Zerinvary Lajos, Dec 05 2007

Mathematica

mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-7*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
k=7; 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)*7^(n/d))); \\ Altug Alkan, Sep 21 2018

Crossrefs

Column 7 of A075195.

Cf. A054614.

Sequence in context: A302523 A290913 A303406 * A220361 A219737 A316106

Adjacent sequences: A054623 A054624 A054625 * A054627 A054628 A054629

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