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%I #48 Jan 15 2021 07:37:39
%S 0,13,28,45,64,85,108,133,160,189,220,253,288,325,364,405,448,493,540,
%T 589,640,693,748,805,864,925,988,1053,1120,1189,1260,1333,1408,1485,
%U 1564,1645,1728,1813,1900,1989,2080,2173,2268,2365,2464,2565,2668,2773
%N a(n) = n*(n + 12).
%H G. C. Greubel, <a href="/A098847/b098847.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hydrogen_spectral_series">Hydrogen spectral series</a>, Humphreys series.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (n+6)^2 - 6^2 = n*(n + 12), n>=0.
%F G.f.: x*(13 - 11*x)/(1-x)^3.
%F a(n) = 2*n + a(n-1) + 11 (with a(0)=0). - _Vincenzo Librandi_, Nov 17 2010
%F 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
%F 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
%F E.g.f.: x*(13 + x)*exp(x). - _G. C. Greubel_, Jul 29 2016
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 18107/332640. - _Amiram Eldar_, Jan 15 2021
%t Table[ n(n + 12), {n, 0, 50}] (* _Robert G. Wilson v_, Jul 14 2005 *)
%t LinearRecurrence[{3,-3,1},{0,13,28},50] (* _Harvey P. Dale_, May 24 2012 *)
%o (PARI) a(n)=n*(n+12) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A098832, A063930.
%Y a(n-6), n>=7, sixth column (used for the n=6 series of the hydrogen atom) of triangle A120070.
%K nonn,easy
%O 0,2
%A Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004
%E More terms from _Robert G. Wilson v_, Jul 14 2005
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