A158680 a(n) = 62*n^2 - 1.

61, 247, 557, 991, 1549, 2231, 3037, 3967, 5021, 6199, 7501, 8927, 10477, 12151, 13949, 15871, 17917, 20087, 22381, 24799, 27341, 30007, 32797, 35711, 38749, 41911, 45197, 48607, 52141, 55799, 59581, 63487, 67517, 71671, 75949, 80351, 84877, 89527, 94301, 99199

Offset

1,1

Comments

The identity (62*n^2 - 1)^2 - (961*n^2 - 31)*(2*n)^2 = 1 can be written as a(n)^2 - A158679(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*(-61 - 64*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 21 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(62))*Pi/sqrt(62))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(62))*Pi/sqrt(62) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {61, 247, 557}, 50] (* Vincenzo Librandi, Feb 19 2012 *)

Prog

(Magma) I:=[61, 247, 557]; [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
(PARI) for(n=1, 40, print1(62*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 19 2012

Crossrefs

Cf. A005843, A158679.

Sequence in context: A251312 A158673 A174333 * A152413 A029815 A142424

Adjacent sequences: A158677 A158678 A158679 * A158681 A158682 A158683

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved