A158736 a(n) = 70*n^2 - 1.

69, 279, 629, 1119, 1749, 2519, 3429, 4479, 5669, 6999, 8469, 10079, 11829, 13719, 15749, 17919, 20229, 22679, 25269, 27999, 30869, 33879, 37029, 40319, 43749, 47319, 51029, 54879, 58869, 62999, 67269, 71679, 76229, 80919, 85749, 90719, 95829, 101079, 106469, 111999

Offset

1,1

Comments

The identity (70*n^2 - 1)^2 - (1225*n^2 - 35)*(2*n)^2 = 1 can be written as a(n)^2 - A158735(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).

Formula

G.f.: x*(-69 - 72*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/sqrt(70))*Pi/sqrt(70))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(70))*Pi/sqrt(70) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {69, 279, 629}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

Prog

(Magma) I:=[69, 279, 629]; [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
(PARI) for(n=1, 40, print1(70*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 20 2012

Crossrefs

Cf. A005843, A158735.

Sequence in context: A188546 A158732 A069216 * A262456 A161486 A236158

Adjacent sequences: A158733 A158734 A158735 * A158737 A158738 A158739

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

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

Status

approved