A098603 a(n) = n*(n+10).

0, 11, 24, 39, 56, 75, 96, 119, 144, 171, 200, 231, 264, 299, 336, 375, 416, 459, 504, 551, 600, 651, 704, 759, 816, 875, 936, 999, 1064, 1131, 1200, 1271, 1344, 1419, 1496, 1575, 1656, 1739, 1824, 1911, 2000, 2091, 2184, 2279, 2376, 2475, 2576, 2679, 2784

Offset: 0

Author

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

Keywords

• nonn
• easy

Comments

These are the only positive integer values of t for which the Binet-de Moivre formula for the recurrence b(n) = 10*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(10^2+4*t) is a positive integer since 10^2+4*t = 10^2+4*a(m) = (2*m+10)^2. Thus the characteristic roots are r1=10+m and r2 = -m. - Felix P. Muga II, Mar 28 2014

Links

• Shawn A. Broyles, Table of n, a(n) for n = 0..1000
• Felix P. 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

Cf. A098832.

a(n-5), n>=6, fifth column (used for the Pfund series of the hydrogen atom) of triangle A120070.

Programs

Magma

[n*(n+10): n in [0..50]]; // G. C. Greubel, Jul 31 2022

Maple

seq(n*(n+10), n=0..53); # Emeric Deutsch, Mar 11 2005

Mathematica

Table[n(n+10),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,24},50] (* Harvey P. Dale, Jul 26 2014 *)

PARI

a(n)=n*(n+10) \\ Charles R Greathouse IV, Jun 16 2017

SageMath

[n*(n+10) for n in (0..50)] # G. C. Greubel, Jul 31 2022

Formula

a(n) = (n+5)^2 - 5^2 = n*(n+10), n>=0.

G.f.: x*(11-9*x)/(1-x)^3.

a(n) = a(n-1) + 2*n + 9, (with a(0)=0). - Vincenzo Librandi, Nov 17 2010

Sum_{n>=1} 1/a(n) = 7381/25200 via sum_{n>=0} 1/((n+x)*(n+y)) = (psi(x)-psi(y))/(x-y). - R. J. Mathar, Jul 14 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=11, a(2)=24. - Harvey P. Dale, Jul 26 2014

Sum_{n>=1} (-1)^(n+1)/a(n) = 1627/25200. - Amiram Eldar, Jan 15 2021

E.g.f.: x*(11 + x)*exp(x). - G. C. Greubel, Jul 31 2022

From Amiram Eldar, Feb 12 2024: (Start)

Product_{n>=1} (1 - 1/a(n)) = -18144*sqrt(2/13)*sin(sqrt(26)*Pi)/(935*Pi).

Product_{n>=1} (1 + 1/a(n)) = 126*sqrt(6)*sin(2*sqrt(6)*Pi)/(23*Pi). (End)

Extensions

More terms from Emeric Deutsch, Mar 11 2005