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

0, 21, 44, 69, 96, 125, 156, 189, 224, 261, 300, 341, 384, 429, 476, 525, 576, 629, 684, 741, 800, 861, 924, 989, 1056, 1125, 1196, 1269, 1344, 1421, 1500, 1581, 1664, 1749, 1836, 1925, 2016, 2109, 2204, 2301, 2400, 2501, 2604, 2709, 2816, 2925, 3036, 3149, 3264

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy

Links

• Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

a(n-10), n >= 11, tenth column (used for the n=10 series of the hydrogen atom) of triangle A120070.

For n*(n+18) see A098850.

Cf. A001008, A001477, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A102928, A120061, A132759, A132760, A132761, A132762, A132765.

Programs

Maple

seq(sum(4*k-n, k=7..n), n=6..51); # Zerinvary Lajos, Feb 17 2008

Mathematica

s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,21,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

PARI

a(n)=n*(n+20) \\ Charles R Greathouse IV, Jan 21 2015

Formula

a(n) = (n+10)^2 - 10^2 = n*(n+20), n >= 0.

G.f.: x*(21-19*x)/(1-x)^3.

a(n) = 2*n + a(n-1) + 19 (with a(0)=0). - Vincenzo Librandi, Nov 13 2010

From Amiram Eldar, Jan 16 2021: (Start)

Sum_{n>=1} 1/a(n) = H(20)/20 = A001008(20)/A102928(20) = 11167027/62078016, where H(k) is the k-th harmonic number.

Sum_{n>=1} (-1)^(n+1)/a(n) = 155685007/4655851200. (End)

From Elmo R. Oliveira, Jan 12 2025: (Start)

E.g.f.: exp(x)*x*(21 + x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)