A158670 a(n) = 60*n^2 - 1.

59, 239, 539, 959, 1499, 2159, 2939, 3839, 4859, 5999, 7259, 8639, 10139, 11759, 13499, 15359, 17339, 19439, 21659, 23999, 26459, 29039, 31739, 34559, 37499, 40559, 43739, 47039, 50459, 53999, 57659, 61439, 65339, 69359, 73499, 77759, 82139, 86639, 91259, 95999

Offset

1,1

Comments

The identity (60*n^2 - 1)^2 - (900*n^2 - 30)*(2*n)^2 = 1 can be written as a(n)^2 - A158669(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*(-59 - 62*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/(2*sqrt(15)))*Pi/(2*sqrt(15)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(15)))*Pi/(2*sqrt(15)) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {59, 239, 539}, 50] (* Vincenzo Librandi, Feb 18 2012 *)

Prog

(Magma) I:=[59, 239, 539]; [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
(PARI) for(n=1, 40, print1(60*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 18 2012

Crossrefs

Cf. A005843, A158669.

Sequence in context: A059256 A158666 A060331 * A142046 A142952 A060263

Adjacent sequences: A158667 A158668 A158669 * A158671 A158672 A158673

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved