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%I #60 Feb 12 2024 02:28:45
%S 0,11,24,39,56,75,96,119,144,171,200,231,264,299,336,375,416,459,504,
%T 551,600,651,704,759,816,875,936,999,1064,1131,1200,1271,1344,1419,
%U 1496,1575,1656,1739,1824,1911,2000,2091,2184,2279,2376,2475,2576,2679,2784
%N a(n) = n*(n+10).
%C These are the only positive integer values of t for which the Binet-de Moivre formula for the recurrence b(n) = 10*b(n-1)+t*b(n-2) with b(0)=0 and b(1)=1 has a root which is a square. In particular, sqrt(10^2+4*t) is a positive integer since 10^2+4*t = 10^2+4*a(m) = (2*m+10)^2. Thus the characteristic roots are r1=10+m and r2 = -m. - _Felix P. Muga II_, Mar 28 2014
%H Shawn A. Broyles, <a href="/A098603/b098603.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>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (n+5)^2 - 5^2 = n*(n+10), n>=0.
%F G.f.: x*(11-9*x)/(1-x)^3.
%F a(n) = a(n-1) + 2*n + 9, (with a(0)=0). - _Vincenzo Librandi_, Nov 17 2010
%F Sum_{n>=1} 1/a(n) = 7381/25200 via sum_{n>=0} 1/((n+x)*(n+y)) = (psi(x)-psi(y))/(x-y). - _R. J. Mathar_, Jul 14 2012
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=11, a(2)=24. - _Harvey P. Dale_, Jul 26 2014
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 1627/25200. - _Amiram Eldar_, Jan 15 2021
%F E.g.f.: x*(11 + x)*exp(x). - _G. C. Greubel_, Jul 31 2022
%F From _Amiram Eldar_, Feb 12 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = -18144*sqrt(2/13)*sin(sqrt(26)*Pi)/(935*Pi).
%F Product_{n>=1} (1 + 1/a(n)) = 126*sqrt(6)*sin(2*sqrt(6)*Pi)/(23*Pi). (End)
%p seq(n*(n+10), n=0..53); # _Emeric Deutsch_, Mar 11 2005
%t Table[n(n+10),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,24},50] (* _Harvey P. Dale_, Jul 26 2014 *)
%o (PARI) a(n)=n*(n+10) \\ _Charles R Greathouse IV_, Jun 16 2017
%o (Magma) [n*(n+10): n in [0..50]]; // _G. C. Greubel_, Jul 31 2022
%o (SageMath) [n*(n+10) for n in (0..50)] # _G. C. Greubel_, Jul 31 2022
%Y Cf. A098832.
%Y a(n-5), n>=6, fifth column (used for the Pfund 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 _Emeric Deutsch_, Mar 11 2005
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