A158764 a(n) = 38*(38*n^2 - 1).

1406, 5738, 12958, 23066, 36062, 51946, 70718, 92378, 116926, 144362, 174686, 207898, 243998, 282986, 324862, 369626, 417278, 467818, 521246, 577562, 636766, 698858, 763838, 831706, 902462, 976106, 1052638, 1132058, 1214366, 1299562, 1387646, 1478618, 1572478

Offset

1,1

Comments

The identity (76*n^2 - 1)^2 - (1444*n^2 - 38)*(2*n)^2 = 1 can be written as A158765(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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*x*(-37-40*x+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>=1} 1/a(n) = (1 - cot(Pi/sqrt(38))*Pi/sqrt(37))/76.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(37))*Pi/sqrt(38) - 1)/76. (End)

From Elmo R. Oliveira, Jan 16 20254: (Start)

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

a(n) = 38*A158596(n). (End)

Mathematica

Table[38 (38 n^2 - 1), {n, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {1406,  5738, 12958}, 40] (* Harvey P. Dale, Jan 09 2012 *)
CoefficientList[Series[38 (- 37 - 40 x + x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)

Prog

(Magma) [38*(38*n^2-1): n in [0..40]]; // Vincenzo Librandi, Sep 11 2013
(PARI) a(n)=38*(38*n^2-1) \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A005843, A158596, A158765.

Sequence in context: A206680 A022058 A107522 * A035863 A045127 A210786

Adjacent sequences: A158761 A158762 A158763 * A158765 A158766 A158767

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 26 2009

Extensions

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved