A158731 a(n) = 1156*n^2 + 34.

34, 1190, 4658, 10438, 18530, 28934, 41650, 56678, 74018, 93670, 115634, 139910, 166498, 195398, 226610, 260134, 295970, 334118, 374578, 417350, 462434, 509830, 559538, 611558, 665890, 722534, 781490, 842758, 906338, 972230, 1040434, 1110950, 1183778, 1258918

Offset

0,1

Comments

The identity (68*n^2 + 1)^2 - (1156*n^2 + 34)*(2*n)^2 = 1 can be written as A158732(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.: -34*(1 + 32*x + 35*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 22 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(34))*Pi/sqrt(34) + 1)/68.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(34))*Pi/sqrt(34) + 1)/68. (End)

From Elmo R. Oliveira, Jan 26 2025: (Start)

E.g.f.: 34*exp(x)*(1 + 34*x + 34*x^2).

a(n) = 34*A158586(n). (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {34, 1190, 4658}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

Prog

(Magma) I:=[34, 1190, 4658]; [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 20 2012
(PARI) for(n=0, 40, print1(1156*n^2 + 34", ")); \\ Vincenzo Librandi, Feb 20 2012

Crossrefs

Cf. A005843, A158586, A158732.

Sequence in context: A189434 A167258 A251924 * A297693 A093550 A296587

Adjacent sequences: A158728 A158729 A158730 * A158732 A158733 A158734

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