A158626 a(n) = 42*n^2 - 1.

41, 167, 377, 671, 1049, 1511, 2057, 2687, 3401, 4199, 5081, 6047, 7097, 8231, 9449, 10751, 12137, 13607, 15161, 16799, 18521, 20327, 22217, 24191, 26249, 28391, 30617, 32927, 35321, 37799, 40361, 43007, 45737, 48551, 51449, 54431, 57497, 60647, 63881, 67199, 70601

Offset

1,1

Comments

The identity (42*n^2-1)^2 - (441*n^2-21)*(2*n)^2 = 1 can be written as a(n)^2-A145678(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 16 2012

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*(-41-44*x+x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 16 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(42))*Pi/sqrt(42))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(42))*Pi/sqrt(42) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {41, 167, 377}, 50] (* Vincenzo Librandi, Feb 16 2012 *)

Prog

(Magma) I:=[41, 167, 377]; [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 16 2012
(PARI) for(n=1, 40, print1(42*n^2-1", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A158625, A145678, A005843.

Sequence in context: A245743 A142839 A142912 * A275269 A142310 A033228

Adjacent sequences: A158623 A158624 A158625 * A158627 A158628 A158629

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 23 2009

Extensions

Edited by R. J. Mathar, Jul 26 2009

A-number updated by R. J. Mathar, Oct 16 2009

Status

approved