A158585 a(n) = 289*n^2 + 17.

17, 306, 1173, 2618, 4641, 7242, 10421, 14178, 18513, 23426, 28917, 34986, 41633, 48858, 56661, 65042, 74001, 83538, 93653, 104346, 115617, 127466, 139893, 152898, 166481, 180642, 195381, 210698, 226593, 243066, 260117, 277746, 295953, 314738, 334101, 354042

Offset

0,1

Comments

The identity (34*n^2 + 1)^2 - (289*n^2 + 17) * (2*n)^2 = 1 can be written as A158586(n)^2 - a(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

G.f.: 17*(1 + 15*x + 18*x^2)/(1-x)^3.

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

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(17))*Pi/sqrt(17) + 1)/34.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(17))*Pi/sqrt(17) + 1)/34. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {17, 306, 1173}, 50] (* Vincenzo Librandi, Feb 15 2012 *)
289*Range[0, 40]^2+17 (* Harvey P. Dale, Dec 26 2019 *)

Prog

(Magma) I:=[17, 306, 1173]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012
(PARI) for(n=0, 50, print1(289*n^2 + 17", ")); \\ Vincenzo Librandi, Feb 15 2012

Crossrefs

Cf. A005843, A158586.

Sequence in context: A083453 A090437 A007805 * A201232 A156085 A129992

Adjacent sequences: A158582 A158583 A158584 * A158586 A158587 A158588

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009

Status

approved