A158558 a(n) = 30*n^2 + 1.

1, 31, 121, 271, 481, 751, 1081, 1471, 1921, 2431, 3001, 3631, 4321, 5071, 5881, 6751, 7681, 8671, 9721, 10831, 12001, 13231, 14521, 15871, 17281, 18751, 20281, 21871, 23521, 25231, 27001, 28831, 30721, 32671, 34681, 36751, 38881, 41071, 43321, 45631, 48001, 50431

Offset: 0

Author

Vincenzo Librandi, Mar 21 2009

Keywords

• nonn
• easy

Comments

The identity (30*n^2+1)^2 - (225*n^2+15)*(2*n)^2 = 1 can be written as a(n)^2 - A158557(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, A158557.

Programs

Magma

I:=[1, 31, 121]; [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}, {1, 31, 121}, 50] (* Vincenzo Librandi, Feb 14 2012 *)
30*Range[0,40]^2+1 (* Harvey P. Dale, Mar 06 2013 *)

PARI

for(n=0, 40, print1(30*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 14 2012

Formula

G.f.: (1 + 28*x + 31*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(30))*Pi/sqrt(30) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(30))*Pi/sqrt(30) + 1)/2. (End)

Extensions

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