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%I #58 Feb 12 2024 02:28:12
%S 0,6,14,24,36,50,66,84,104,126,150,176,204,234,266,300,336,374,414,
%T 456,500,546,594,644,696,750,806,864,924,986,1050,1116,1184,1254,1326,
%U 1400,1476,1554,1634,1716,1800,1886,1974,2064,2156
%N a(n) = n*(n+5).
%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)=5*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(5^2+4*t) is an integer since 5^2+4*t=5^2+4*a(m)=(2*m+5)^2. Thus the characteristic roots are r1=m+5 and r2=-m. - _Felix P. Muga II_, Mar 27 2014
%H Shawn A. Broyles, <a href="/A028557/b028557.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 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 Aleksandar Petojević, <a href="http://dx.doi.org/10.5937/MatMor0801037P">A Note about the Pochhammer Symbol</a>, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 2*A055998(n).
%F a(n) = 2*n+a(n-1)+4. - _Vincenzo Librandi_, Aug 05 2010
%F Sum_{n>=1} 1/a(n) = 137/300 = 0.4566666... - _R. J. Mathar_, Mar 22 2011
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/5 - 47/300. - _Amiram Eldar_, Jan 15 2021
%F From _Amiram Eldar_, Feb 12 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = -24*cos(sqrt(29)*Pi/2)/(7*Pi).
%F Product_{n>=1} (1 + 1/a(n)) = 8*cos(sqrt(21)*Pi/2)/Pi. (End)
%t Table[n(n+5),{n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, May 19 2011 *)
%o (PARI) a(n)=n*(n+5) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A002522, A055998, A104675.
%K nonn,easy
%O 0,2
%A _Patrick De Geest_
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