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%I #35 Sep 08 2022 08:45:01
%S 0,1,20,315,2344,11165,39996,117655,299600,683289,1428580,2783891,
%T 5118840,8964085,15059084,24408495,38347936,58619825,87460020,
%U 127695979,182857160,257298381,356336860,486403655,655210224,871930825,1147401476,1494336195,1927561240
%N Number of ways to color vertices of a heptagon using <= n colors, allowing only rotations.
%C Length-7 necklaces with n kinds of beads. - _Vincenzo Librandi_ Apr 30 2012
%H Vincenzo Librandi, <a href="/A054621/b054621.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F a(n) = (1/7) * Sum_{d|7} phi(d)*n^(7/d) = (1/7) * (n^7 + 6*n). [corrected by _Klaus Wagner_, Sep 15 2015]
%F G.f.: x*(1+12*x+183*x^2+328*x^3+183*x^4+12*x^5+x^6) / (1-x)^8. - _R. J. Mathar_, Aug 30 2011
%F a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8), n>7. - _Wesley Ivan Hurt_, Sep 15 2015
%p a:=proc(n) option remember:
%p if n=0 then 0 elif n=1 then 1 elif n=2 then 20 elif n=3 then 315 elif n=4 then 2344 elif n=5 then 11165 elif n=6 then 39996 elif n=7 then 117655 elif n=8 then 299600 else
%p 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8): fi: end: seq(a(n), n=0..50); # _Wesley Ivan Hurt_, Sep 15 2015
%t CoefficientList[Series[x*(1+12*x+183*x^2+328*x^3+183*x^4+ 12*x^5+x^6)/(x-1)^8,{x,0,33}],x] (* _Vincenzo Librandi_, Apr 30 2012 *)
%o (Magma) I:=[0, 1, 20, 315, 2344, 11165, 39996, 117655]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5)-28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..30]]; // _Vincenzo Librandi_, Apr 30 2012
%Y Cf. A054622. Row 7 of A075195.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Apr 16 2000
%E Edited by _Christian G. Bower_, Sep 07 2002
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