A158672 a(n) = 900*n^2 + 30.

30, 930, 3630, 8130, 14430, 22530, 32430, 44130, 57630, 72930, 90030, 108930, 129630, 152130, 176430, 202530, 230430, 260130, 291630, 324930, 360030, 396930, 435630, 476130, 518430, 562530, 608430, 656130, 705630, 756930, 810030, 864930, 921630, 980130, 1040430

Offset: 0

Author

Vincenzo Librandi, Mar 24 2009

Keywords

• nonn
• easy

Comments

The identity (60*n^2 + 1)^2 - (900*n^2 + 30)*(2*n)^2 = 1 can be written as A158673(n)^2 - a(n)*A005843(n)^2 = 1.

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, A158558, A158673.

Programs

Magma

I:=[30, 930, 3630]; [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 19 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {30, 930, 3630}, 50] (* Vincenzo Librandi, Feb 19 2012 *)
900*Range[0,40]^2+30 (* Harvey P. Dale, May 02 2025 *)

PARI

for(n=0, 40, print1(900*n^2 + 30", ")); \\ Vincenzo Librandi, Feb 19 2012

Formula

G.f.: -30*(1 + 28*x + 31*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 20 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(30))*Pi/sqrt(30) + 1)/60.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(30))*Pi/sqrt(30) + 1)/60. (End)

From Elmo R. Oliveira, Jan 15 2025: (Start)

E.g.f.: 30*exp(x)*(1 + 30*x + 30*x^2).

a(n) = 30*A158558(n). (End)

Extensions

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009