A158732 a(n) = 68*n^2 + 1.

1, 69, 273, 613, 1089, 1701, 2449, 3333, 4353, 5509, 6801, 8229, 9793, 11493, 13329, 15301, 17409, 19653, 22033, 24549, 27201, 29989, 32913, 35973, 39169, 42501, 45969, 49573, 53313, 57189, 61201, 65349, 69633, 74053, 78609, 83301, 88129, 93093, 98193, 103429

Offset

0,2

Comments

The identity (68*n^2 + 1)^2 - (1156*n^2 + 34)*(2*n)^2 = 1 can be written as a(n)^2 - A158731(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..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

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

G.f.: -(1 + 66*x + 69*x^2)/(x-1)^3.

From Amiram Eldar, Mar 22 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(17)))*Pi/(2*sqrt(17)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(17)))*Pi/(2*sqrt(17)) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 69, 273}, 50] (* Vincenzo Librandi, Feb 18 2012 *)
68*Range[0, 40]^2+1 (* Harvey P. Dale, Sep 16 2019 *)

Prog

MAGMA) I:=[1, 69, 273]; [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=0, 40, print1(68*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 18 2012

Crossrefs

Cf. A005843, A158732.

Sequence in context: A183447 A296126 A188546 * A069216 A158736 A262456

Adjacent sequences: A158729 A158730 A158731 * A158733 A158734 A158735

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved