OFFSET
0,2
COMMENTS
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
LINKS
Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), pp. 37-42.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 2*A055998(n).
a(n) = 2*n + a(n-1) + 4. - Vincenzo Librandi, Aug 05 2010
Sum_{n>=1} 1/a(n) = 137/300 = 0.4566666... - R. J. Mathar, Mar 22 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/5 - 47/300. - Amiram Eldar, Jan 15 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -24*cos(sqrt(29)*Pi/2)/(7*Pi).
Product_{n>=1} (1 + 1/a(n)) = 8*cos(sqrt(21)*Pi/2)/Pi. (End)
From Elmo R. Oliveira, Oct 28 2024: (Start)
G.f.: 2*x*(3 - 2*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(6 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
MATHEMATICA
Table[n(n+5), {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)
PROG
(PARI) a(n)=n*(n+5) \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved