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A054622 Number of ways to color vertices of an octagon using <= n colors, allowing only rotations.

%I A054622 #30 Sep 08 2022 08:45:01
%S A054622 0,1,36,834,8230,48915,210126,720916,2097684,5381685,12501280,
%T A054622 26796726,53750346,101969959,184478490,320367720,536879176,871980201,
%U A054622 1377508284,2122961770,3200020110,4727881851,6859513606,9788908284,13759455900
%N A054622 Number of ways to color vertices of an octagon using <= n colors, allowing only rotations.
%C A054622 Length-8 necklaces with n kinds of beads. - _Joerg Arndt_, Apr 29 2012
%H A054622 Vincenzo Librandi, <a href="/A054622/b054622.txt">Table of n, a(n) for n = 0..1000</a>
%H A054622 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A054622 a(n) = Sum_{d|8} phi(d)*n^(8/d)/8 = n*(n+1)*(n^6-n^5+n^4-n^3+2*n^2-2*n+4)/8.
%F A054622 G.f.: x*(1+27*x+546*x^2+1936*x^3+1971*x^4+525*x^5+34*x^6)/(1-x)^9. - _Colin Barker_, Jan 29 2012
%F A054622 a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). _Vincenzo Librandi_, Apr 29 2012
%t A054622 CoefficientList[Series[x*(1+27*x+546*x^2+1936*x^3+ 1971*x^4+525*x^5+34*x^6)/(1-x)^9,{x,0,30}],x] (* _Vincenzo Librandi_, Apr 29 2012 *)
%o A054622 (Magma) I:=[0, 1, 36, 834, 8230, 48915, 210126, 720916, 2097684]; [n le 9 select I[n] else 9*Self(n-1)-36*Self(n-2)+84*Self(n-3)-126*Self(n-4)+126*Self(n-5)-84*Self(n-6)+36*Self(n-7)-9*Self(n-8)+Self(n-9): n in [1..30]]; // _Vincenzo Librandi_, Apr 29 2012
%Y A054622 Row 8 of A075195.
%Y A054622 Cf. A064621, A054623, ...
%K A054622 nonn,easy
%O A054622 0,3
%A A054622 _N. J. A. Sloane_, Apr 16 2000
%E A054622 Edited by _Christian G. Bower_, Sep 07 2002