A098847 a(n) = n*(n + 12).

0, 13, 28, 45, 64, 85, 108, 133, 160, 189, 220, 253, 288, 325, 364, 405, 448, 493, 540, 589, 640, 693, 748, 805, 864, 925, 988, 1053, 1120, 1189, 1260, 1333, 1408, 1485, 1564, 1645, 1728, 1813, 1900, 1989, 2080, 2173, 2268, 2365, 2464, 2565, 2668, 2773

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.

Wikipedia, Hydrogen spectral series, Humphreys series.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = (n+6)^2 - 6^2 = n*(n + 12), n>=0.

G.f.: x*(13 - 11*x)/(1-x)^3.

a(n) = 2*n + a(n-1) + 11 (with a(0)=0). - Vincenzo Librandi, Nov 17 2010

a(0)=0, a(1)=13, a(2)=28, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 24 2012

Sum_{n>=1} 1/a(n) = 86021/332640 = 0.258600... via Sum_{n>=0} 1/((n+x)(n+y)) = (psi(x)-psi(y))/(x-y). - R. J. Mathar, Jul 14 2012

E.g.f.: x*(13 + x)*exp(x). - G. C. Greubel, Jul 29 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = 18107/332640. - Amiram Eldar, Jan 15 2021

Mathematica

Table[ n(n + 12), {n, 0, 50}] (* Robert G. Wilson v, Jul 14 2005 *)
LinearRecurrence[{3, -3, 1}, {0, 13, 28}, 50] (* Harvey P. Dale, May 24 2012 *)

Prog

(PARI) a(n)=n*(n+12) \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A098832, A063930.

a(n-6), n>=7, sixth column (used for the n=6 series of the hydrogen atom) of triangle A120070.

Sequence in context: A046044 A026919 A063309 * A161453 A038597 A026054

Adjacent sequences: A098844 A098845 A098846 * A098848 A098849 A098850

Keyword

nonn,easy

Author

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

Extensions

More terms from Robert G. Wilson v, Jul 14 2005

Status

approved