A158730 a(n) = 68*n^2 - 1.

67, 271, 611, 1087, 1699, 2447, 3331, 4351, 5507, 6799, 8227, 9791, 11491, 13327, 15299, 17407, 19651, 22031, 24547, 27199, 29987, 32911, 35971, 39167, 42499, 45967, 49571, 53311, 57187, 61199, 65347, 69631, 74051, 78607, 83299, 88127, 93091, 98191, 103427, 108799

Offset: 1

Author

Vincenzo Librandi, Mar 25 2009

Keywords

• nonn
• easy

Comments

The identity (68*n^2 - 1)^2 - (1156*n^2 - 34)*(2*n)^2 = 1 can be written as a(n)^2 - A158729(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, A141759, A158729.

Programs

Magma

I:=[67, 271, 611]; [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 20 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {67, 271, 611}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

PARI

for(n=1, 40, print1(68*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 20 2012

Formula

G.f.: x*(-67 - 70*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 22 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(17)))*Pi/(2*sqrt(17)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(17)))*Pi/(2*sqrt(17)) - 1)/2. (End)

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

E.g.f.: exp(x)*(68*x^2 + 68*x - 1) + 1.

a(n) = A141759(2*n). (End)

Extensions

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009