A158492 a(n) = 100*n^2 + 10.

10, 110, 410, 910, 1610, 2510, 3610, 4910, 6410, 8110, 10010, 12110, 14410, 16910, 19610, 22510, 25610, 28910, 32410, 36110, 40010, 44110, 48410, 52910, 57610, 62510, 67610, 72910, 78410, 84110, 90010, 96110, 102410, 108910, 115610, 122510, 129610, 136910, 144410

Offset: 0

Author

Vincenzo Librandi, Mar 21 2009

Keywords

• nonn
• easy

Comments

The identity (20*n^2 + 1)^2 - (100*n^2 + 10)*(2*n)^2 = 1 can be written as A158493(n)^2 - a(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 21 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A158187, A158493.

Programs

Magma

I:=[10, 110, 410]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 21 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {10, 110, 410}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
100Range[0,40]^2+10 (* Harvey P. Dale, Dec 30 2019 *)

PARI

for(n=0, 40, print1(100*n^2 + 10", ")); \\ Vincenzo Librandi, Feb 21 2012

Formula

From Vincenzo Librandi, Feb 21 2012: (Start)

G.f.: -(10 + 80*x + 110*x^2)/(x-1)^3;

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

From Amiram Eldar, Mar 05 2023: (Start)

Sum_{n>=0} 1/a(n) = (1 + coth(Pi/sqrt(10))*Pi/sqrt(10))/20.

Sum_{n>=0} (-1)^n/a(n) = (1 + cosech(Pi/sqrt(10))*Pi/sqrt(10))/20. (End)

From Elmo R. Oliveira, Jan 17 2025: (Start)

E.g.f.: 10*exp(x)*(1 + 10*x + 10*x^2).

a(n) = 10*A158187(n). (End)

Extensions

Edited by N. J. A. Sloane, Oct 12 2009