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A120071 a(n) = n*(n+20).

%I A120071 #31 Jan 12 2025 23:36:58
%S A120071 0,21,44,69,96,125,156,189,224,261,300,341,384,429,476,525,576,629,
%T A120071 684,741,800,861,924,989,1056,1125,1196,1269,1344,1421,1500,1581,1664,
%U A120071 1749,1836,1925,2016,2109,2204,2301,2400,2501,2604,2709,2816,2925,3036,3149,3264
%N A120071 a(n) = n*(n+20).
%H A120071 Felix P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, Preprint on ResearchGate, March 2014.
%H A120071 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A120071 a(n) = (n+10)^2 - 10^2 = n*(n+20), n >= 0.
%F A120071 G.f.: x*(21-19*x)/(1-x)^3.
%F A120071 a(n) = 2*n + a(n-1) + 19 (with a(0)=0). - _Vincenzo Librandi_, Nov 13 2010
%F A120071 From _Amiram Eldar_, Jan 16 2021: (Start)
%F A120071 Sum_{n>=1} 1/a(n) = H(20)/20 = A001008(20)/A102928(20) = 11167027/62078016, where H(k) is the k-th harmonic number.
%F A120071 Sum_{n>=1} (-1)^(n+1)/a(n) = 155685007/4655851200. (End)
%F A120071 From _Elmo R. Oliveira_, Jan 12 2025: (Start)
%F A120071 E.g.f.: exp(x)*x*(21 + x).
%F A120071 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)
%p A120071 seq(sum(4*k-n, k=7..n), n=6..51); # _Zerinvary Lajos_, Feb 17 2008
%t A120071 s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,21,6!,2}];lst (* _Vladimir Joseph Stephan Orlovsky_, Feb 26 2009 *)
%o A120071 (PARI) a(n)=n*(n+20) \\ _Charles R Greathouse IV_, Jan 21 2015
%Y A120071 a(n-10), n >= 11, tenth column (used for the n=10 series of the hydrogen atom) of triangle A120070.
%Y A120071 For n*(n+18) see A098850.
%Y A120071 Cf. A001008, A001477, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A102928, A120061, A132759, A132760, A132761, A132762, A132765.
%K A120071 nonn,easy
%O A120071 0,2
%A A120071 _Wolfdieter Lang_, Jul 20 2006