A158669 a(n) = 900*n^2 - 30.

870, 3570, 8070, 14370, 22470, 32370, 44070, 57570, 72870, 89970, 108870, 129570, 152070, 176370, 202470, 230370, 260070, 291570, 324870, 359970, 396870, 435570, 476070, 518370, 562470, 608370, 656070, 705570, 756870, 809970, 864870, 921570, 980070, 1040370, 1102470

Offset: 1

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 A158670(n)^2 - a(n)*A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..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, A158560, A158670.

Programs

Magma

I:=[870, 3570, 8070]; [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 18 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {870, 3570, 8070}, 50] (* Vincenzo Librandi, Feb 18 2012 *)

PARI

for(n=1, 40, print1(900*n^2 - 30", ")); \\ Vincenzo Librandi, Feb 18 2012

Formula

G.f.: 30*x*(-29 - 32*x + 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>=1} 1/a(n) = (1 - cot(Pi/sqrt(30))*Pi/sqrt(30))/60.

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

From Elmo R. Oliveira, Jan 16 2025: (Start)

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

a(n) = 30*A158560(n). (End)

Extensions

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009