A158597 a(n) = 400*n^2 - 20.

380, 1580, 3580, 6380, 9980, 14380, 19580, 25580, 32380, 39980, 48380, 57580, 67580, 78380, 89980, 102380, 115580, 129580, 144380, 159980, 176380, 193580, 211580, 230380, 249980, 270380, 291580, 313580, 336380, 359980, 384380, 409580, 435580, 462380, 489980, 518380

Offset

1,1

Comments

The identity (40*n^2 - 1)^2 - (400*n^2 - 20)*(2*n)^2 = 1 can be written as A158598(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..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.: 20*x*(-19 - 22*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)))/40.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) - 1)/40. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {380, 1580, 3580}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
400*Range[40]^2-20 (* Harvey P. Dale, Nov 04 2015 *)

Prog

(Magma) I:=[380, 1580, 3580]; [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 16 2012
(PARI) for(n=1, 40, print1(400*n^2- 20", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A005843, A158598.

Sequence in context: A206349 A252130 A252123 * A252122 A250635 A233952

Adjacent sequences: A158594 A158595 A158596 * A158598 A158599 A158600

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

Status

approved