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%I #31 Sep 08 2022 08:46:13
%S 0,76,2786,46764,439204,2744420,12813606,48229636,153992264,432083484,
%T 1092730090,2537720636,5489037036,11179326964,21624372014,40001698260,
%U 71163830416,122319408236,203920464114,330799604044,523606640180,810600392196,1229857906486
%N a(n) = n*(n^2 + 3)*(n^6 + 6*n^4 + 9*n^2 + 3).
%C Also numbers of the form (n-th metallic mean)^9 - 1/(n-th metallic mean)^9, see link to Wikipedia.
%H Colin Barker, <a href="/A261574/b261574.txt">Table of n, a(n) for n = 0..1000</a> (first 100 terms from Raphael Ranna)
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Metallic_mean">Metallic mean</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F a(n) = -a(-n) = ( (n+sqrt(n^2+4))/2 )^9-1/( (n+sqrt(n^2+4))/2 )^9.
%F G.f.: 2*x*(38*x^8 +1013*x^7 +11162*x^6 +43907*x^5 +69200*x^4 +43907*x^3 +11162*x^2 +1013*x +38) / (x -1)^10. - _Colin Barker_, Aug 25 2015
%t Table[n (n^2 + 3) (n^6 + 6 n^4 + 9 n^2 + 3), {n, 0, 25}] (* _Bruno Berselli_, Aug 25 2015 *)
%o (PARI) concat(0, Vec(2*x*(38*x^8 +1013*x^7 +11162*x^6 +43907*x^5 +69200*x^4 +43907*x^3 +11162*x^2 +1013*x +38) / (x -1)^10 + O(x^50))) \\ _Colin Barker_, Aug 25 2015
%o (Magma) [n*(n^2+3)*(n^6+6*n^4+9*n^2+3): n in [0..25]]; // _Bruno Berselli_, Aug 25 2015
%Y Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458, A176522, A261391, A261540.
%K nonn,easy
%O 0,2
%A _Raphael Ranna_, Aug 24 2015
%E Formula in Name by _Colin Barker_, Aug 25 2015
%E Offset changed from 1 to 0 and initial 0 added by _Bruno Berselli_, Aug 25 2015
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