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%I #64 Feb 12 2024 02:28:25
%S 0,9,20,33,48,65,84,105,128,153,180,209,240,273,308,345,384,425,468,
%T 513,560,609,660,713,768,825,884,945,1008,1073,1140,1209,1280,1353,
%U 1428,1505,1584,1665,1748,1833,1920,2009,2100,2193,2288,2385
%N a(n) = n*(n+8).
%C a(m) where m is a positive integer are the only positive integer values of t for which the Binet-de Moivre Formula of the recurrence b(n) = 8*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(8^2 + 4*t) is a positive integer since 8^2 + 4*t = 8^2 + 4*a(m) = (2*m + 8)^2. Thus, the characteristics roots are r1 = 8 + m and r2 = -m. - _Felix P. Muga II_, Mar 28 2014
%H Shawn A. Broyles, <a href="/A028566/b028566.txt">Table of n, a(n) for n = 0..1000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</a>.
%H Felix Pozon 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+4)^2 - 4^2 = n*(n+8), n >= 0.
%F G.f.: x*(9 - 7*x)/(1 - x)^3.
%F a(n) = 2*n + a(n-1) + 7. - _Vincenzo Librandi_, Aug 05 2010
%F Sum_{n >= 1} 1/a(n) = 761/2240 = 0.3397321... - _R. J. Mathar_, Mar 22 2011
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 533/6720. - _Amiram Eldar_, Jan 15 2021
%F E.g.f.: x*(9 + 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)) = 315*sin(sqrt(17)*Pi)/(13*sqrt(17)*Pi).
%F Product_{n>=1} (1 + 1/a(n)) = -32*sqrt(15)*sin(sqrt(15)*Pi)/(11*Pi). (End)
%t Table[n (n+8), {n, 0, 50}] (* _Bruno Berselli_, Apr 06 2014 *)
%o (PARI) a(n)=n*(n+8)
%o (Sage) [n*(n+8) for n in [0..50]] # _Bruno Berselli_, Apr 06 2014
%o (Scala) (0 to 49).map { n: Int => n * n + 8 * n } // _Alonso del Arte_, Nov 10 2019
%o (Magma) [n*(n+8): n in [0..50]]; // _G. C. Greubel_, Jul 31 2022
%Y a(n - 4), n >= 5, fourth column (used for the Brackett series of the hydrogen atom) of triangle A120070.
%K nonn,easy
%O 0,2
%A _Patrick De Geest_
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