A054623 Number of ways to color vertices of a 9-gon using <= n colors, allowing only rotations.

0, 1, 60, 2195, 29144, 217045, 1119796, 4483815, 14913200, 43046889, 111111340, 261994491, 573309320, 1178278205, 2295672484, 4271485135, 7635498336, 13176431825, 22039922460, 35854190179, 56888890680, 88253340581, 134141026580

Offset: 0

Author

N. J. A. Sloane, Apr 16 2000

Keywords

• nonn
• easy

Comments

Length-9 necklaces with n kinds of beads. [Vincenzo Librandi, Apr 29 2012]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Row 9 of A075195.

Programs

Magma

[n*(n^8+2*n^2+6)/9: n in [0..30]]; // Vincenzo Librandi, Apr 30 2012

Mathematica

CoefficientList[Series[x*(1+50*x+1640*x^2+9774*x^3+17390*x^4+9774*x^5+1640*x^6+50*x^7+x^8)/(1-x)^10,{x,0,30}],x] (* Vincenzo Librandi, Apr 29 2012 *)

Formula

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

a(n) = n*(n^8+2*n^2+6)/9.

G.f.: x*(1+50*x+1640*x^2+9774*x^3+17390*x^4+9774*x^5+1640*x^6+50*x^7+x^8)/ (1-x)^10. [Colin Barker, Jan 29 2012]

a(n) = 10*a(n-1) -45*a(n-2) +120*a(n-3) -210*a(n-4)+252*a(n-5) -210*a(n-6) +120*a(n-7) -45*a(n-8) +10*a(n-9) -a(n-10). [Vincenzo Librandi, Apr 29 2012]

Extensions

Edited by Christian G. Bower, Sep 07 2002