A158548 a(n) = 169*n^2 + 13.

13, 182, 689, 1534, 2717, 4238, 6097, 8294, 10829, 13702, 16913, 20462, 24349, 28574, 33137, 38038, 43277, 48854, 54769, 61022, 67613, 74542, 81809, 89414, 97357, 105638, 114257, 123214, 132509, 142142, 152113, 162422, 173069, 184054, 195377, 207038, 219037, 231374

Offset

0,1

Comments

The identity (26*n^2 + 1)^2 - (169*n^2 + 13)*(2*n)^2 = 1 can be written as A158549(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f: 13*(1 + 11*x + 14*x^2)/(1-x)^3.

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

From Amiram Eldar, Mar 06 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(13))*Pi/sqrt(13) + 1)/26.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(13))*Pi/sqrt(13) + 1)/26. (End)

E.g.f.: 13*exp(x)*(1 + 13*x + 13*x^2). - Elmo R. Oliveira, Jan 15 2025

Mathematica

LinearRecurrence[{3, -3, 1}, {13, 182, 689}, 50] (* Vincenzo Librandi, Feb 14 2012 *)

Prog

(Magma) I:=[13, 182, 689]; [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 14 2012
(PARI) for(n=1, 40, print1(169*n^2 + 13", ")); \\ Vincenzo Librandi, Feb 14 2012

Crossrefs

Cf. A005843, A158549.

Sequence in context: A067385 A097260 A178303 * A285399 A297581 A268413

Adjacent sequences: A158545 A158546 A158547 * A158549 A158550 A158551

Keyword

nonn,easy,changed

Author

Vincenzo Librandi, Mar 21 2009

Extensions

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

Status

approved