A028557 a(n) = n*(n+5).

0, 6, 14, 24, 36, 50, 66, 84, 104, 126, 150, 176, 204, 234, 266, 300, 336, 374, 414, 456, 500, 546, 594, 644, 696, 750, 806, 864, 924, 986, 1050, 1116, 1184, 1254, 1326, 1400, 1476, 1554, 1634, 1716, 1800, 1886, 1974, 2064, 2156, 2250, 2346, 2444, 2544, 2646, 2750

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A002522, A055998, A104675.

Programs

Mathematica

Table[n(n+5),{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)

PARI

a(n)=n*(n+5) \\ Charles R Greathouse IV, Jun 17 2017

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)