This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A028557 #61 Oct 28 2024 19:39:26
%S A028557 0,6,14,24,36,50,66,84,104,126,150,176,204,234,266,300,336,374,414,
%T A028557 456,500,546,594,644,696,750,806,864,924,986,1050,1116,1184,1254,1326,
%U A028557 1400,1476,1554,1634,1716,1800,1886,1974,2064,2156,2250,2346,2444,2544,2646,2750
%N A028557 a(n) = n*(n+5).
%C A028557 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 A028557 Shawn A. Broyles, <a href="/A028557/b028557.txt">Table of n, a(n) for n = 0..1000</a>
%H A028557 Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</a>.
%H A028557 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 A028557 Aleksandar Petojević, <a href="http://dx.doi.org/10.5937/MatMor0801037P">A Note about the Pochhammer Symbol</a>, Mathematica Moravica, Vol. 12-1 (2008), pp. 37-42.
%H A028557 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A028557 a(n) = 2*A055998(n).
%F A028557 a(n) = 2*n + a(n-1) + 4. - _Vincenzo Librandi_, Aug 05 2010
%F A028557 Sum_{n>=1} 1/a(n) = 137/300 = 0.4566666... - _R. J. Mathar_, Mar 22 2011
%F A028557 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/5 - 47/300. - _Amiram Eldar_, Jan 15 2021
%F A028557 From _Amiram Eldar_, Feb 12 2024: (Start)
%F A028557 Product_{n>=1} (1 - 1/a(n)) = -24*cos(sqrt(29)*Pi/2)/(7*Pi).
%F A028557 Product_{n>=1} (1 + 1/a(n)) = 8*cos(sqrt(21)*Pi/2)/Pi. (End)
%F A028557 From _Elmo R. Oliveira_, Oct 28 2024: (Start)
%F A028557 G.f.: 2*x*(3 - 2*x)/(1 - x)^3.
%F A028557 E.g.f.: exp(x)*x*(6 + x).
%F A028557 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A028557 Table[n(n+5),{n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, May 19 2011 *)
%o A028557 (PARI) a(n)=n*(n+5) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A028557 Cf. A002522, A055998, A104675.
%K A028557 nonn,easy
%O A028557 0,2
%A A028557 _Patrick De Geest_