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

%I A054623 #30 Sep 08 2022 08:45:01
%S A054623 0,1,60,2195,29144,217045,1119796,4483815,14913200,43046889,111111340,
%T A054623 261994491,573309320,1178278205,2295672484,4271485135,7635498336,
%U A054623 13176431825,22039922460,35854190179,56888890680,88253340581,134141026580
%N A054623 Number of ways to color vertices of a 9-gon using <= n colors, allowing only rotations.
%C A054623 Length-9 necklaces with n kinds of beads. [_Vincenzo Librandi_, Apr 29 2012]
%H A054623 Vincenzo Librandi, <a href="/A054623/b054623.txt">Table of n, a(n) for n = 0..1000</a>
%H A054623 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A054623 a(n) = Sum_{d|9} phi(d)*n^(9/d)/9.
%F A054623 a(n) = n*(n^8+2*n^2+6)/9.
%F A054623 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]
%F A054623 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]
%t A054623 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 *)
%o A054623 (Magma) [n*(n^8+2*n^2+6)/9: n in [0..30]]; // _Vincenzo Librandi_, Apr 30 2012
%Y A054623 Row 9 of A075195.
%K A054623 nonn,easy
%O A054623 0,3
%A A054623 _N. J. A. Sloane_, Apr 16 2000
%E A054623 Edited by _Christian G. Bower_, Sep 07 2002