A028566 a(n) = n*(n+8).

0, 9, 20, 33, 48, 65, 84, 105, 128, 153, 180, 209, 240, 273, 308, 345, 384, 425, 468, 513, 560, 609, 660, 713, 768, 825, 884, 945, 1008, 1073, 1140, 1209, 1280, 1353, 1428, 1505, 1584, 1665, 1748, 1833, 1920, 2009, 2100, 2193, 2288, 2385

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) = 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

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 Pozon Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
• Wikipedia, Hydrogen spectral series.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

a(n - 4), n >= 5, fourth column (used for the Brackett series of the hydrogen atom) of triangle A120070.

Programs

Magma

[n*(n+8): n in [0..50]]; // G. C. Greubel, Jul 31 2022

Mathematica

Table[n (n+8), {n, 0, 50}] (* Bruno Berselli, Apr 06 2014 *)

PARI

a(n)=n*(n+8)

Sage

[n*(n+8) for n in [0..50]] # Bruno Berselli, Apr 06 2014

Scala

(0 to 49).map { n: Int => n * n + 8 * n } // Alonso del Arte, Nov 10 2019

Formula

a(n) = (n+4)^2 - 4^2 = n*(n+8), n >= 0.

G.f.: x*(9 - 7*x)/(1 - x)^3.

a(n) = 2*n + a(n-1) + 7. - Vincenzo Librandi, Aug 05 2010

Sum_{n >= 1} 1/a(n) = 761/2240 = 0.3397321... - R. J. Mathar, Mar 22 2011

Sum_{n>=1} (-1)^(n+1)/a(n) = 533/6720. - Amiram Eldar, Jan 15 2021

E.g.f.: x*(9 + x)*exp(x). - G. C. Greubel, Jul 31 2022

From Amiram Eldar, Feb 12 2024: (Start)

Product_{n>=1} (1 - 1/a(n)) = 315*sin(sqrt(17)*Pi)/(13*sqrt(17)*Pi).

Product_{n>=1} (1 + 1/a(n)) = -32*sqrt(15)*sin(sqrt(15)*Pi)/(11*Pi). (End)