A158766 a(n) = 1444*n^2 + 38.

38, 1482, 5814, 13034, 23142, 36138, 52022, 70794, 92454, 117002, 144438, 174762, 207974, 244074, 283062, 324938, 369702, 417354, 467894, 521322, 577638, 636842, 698934, 763914, 831782, 902538, 976182, 1052714, 1132134, 1214442, 1299638, 1387722, 1478694, 1572554

Offset

0,1

Comments

The identity (76*n^2 + 1)^2 - (1444*n^2 + 38)*(2*n)^2 = 1 can be written as A158767(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.: -38*(1 + 36*x + 39*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 23 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(38))*Pi/sqrt(38) + 1)/76.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(38))*Pi/sqrt(38) + 1)/76. (End)

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

E.g.f.: 38*exp(x)*(1 + 38*x + 38*x^2).

a(n) = 38*A158593(n). (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {38, 1482, 5814}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

Prog

(Magma) I:=[38, 1482, 5814]; [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 21 2012
(PARI) for(n=0, 40, print1(1444*n^2 + 38", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158593, A158767.

Sequence in context: A009982 A041685 A221385 * A217224 A272036 A055605

Adjacent sequences: A158763 A158764 A158765 * A158767 A158768 A158769

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 26 2009

Extensions

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

Status

approved