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
Keywords
Examples
G.f. = 1 + 7*x + 28*x^2 + 119*x^3 + 616*x^4 + 3367*x^5 + 19684*x^6 + ...
Links
- Eric Weisstein's World of Mathematics, Necklace.
- Index entries for sequences related to necklaces
Programs
-
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 *) -
PARI
a(n)=if(n==0, 1, 1/n*sumdiv(n, d, eulerphi(d)*7^(n/d))); \\ Altug Alkan, Sep 21 2018
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
Extensions
Edited by Christian G. Bower, Sep 07 2002
a(0) corrected by Herbert Kociemba, Nov 02 2016