A158555 a(n) = 196*n^2 + 14.

14, 210, 798, 1778, 3150, 4914, 7070, 9618, 12558, 15890, 19614, 23730, 28238, 33138, 38430, 44114, 50190, 56658, 63518, 70770, 78414, 86450, 94878, 103698, 112910, 122514, 132510, 142898, 153678, 164850, 176414, 188370, 200718, 213458, 226590, 240114, 254030

Offset: 0

Author

Vincenzo Librandi, Mar 21 2009

Keywords

• nonn
• easy

Comments

The identity (28*n^2 + 1)^2 -(196*n^2 + 14)*(2*n)^2 = 1 can be written as A158556(n)^2 - a(n)*A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A158556.

Programs

Magma

I:=[14, 210, 798]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 14 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {14, 210, 798}, 50] (* Vincenzo Librandi, Feb 05 2012 *)

PARI

for(n=0, 40, print1(196*n^2 + 14", ")); \\ Vincenzo Librandi, Feb 14 2012

Formula

G.f.: 14*(1 + 12*x + 15*x^2)/(1-x)^3.

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

From Amiram Eldar, Mar 09 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(14))*Pi/sqrt(14) + 1)/28.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(14))*Pi/sqrt(14) + 1)/28. (End)

E.g.f.: 14*exp(x)*(1 + 14*x + 14*x^2). - Elmo R. Oliveira, Jan 15 2025

Extensions

Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009