A158683 a(n) = 1024*n^2 - 32.

992, 4064, 9184, 16352, 25568, 36832, 50144, 65504, 82912, 102368, 123872, 147424, 173024, 200672, 230368, 262112, 295904, 331744, 369632, 409568, 451552, 495584, 541664, 589792, 639968, 692192, 746464, 802784, 861152, 921568, 984032, 1048544, 1115104, 1183712

Offset

1,1

Comments

The identity (64*n^2 - 1)^2 - (1024*n^2 - 32)*(2*n)^2 = 1 can be written as A158684(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.: 32*x*(-31-34*x+x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 21 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/64.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) - 1)/64. (End)

From Elmo R. Oliveira, Jan 16 2025: (Start)

E.g.f.: 32*(exp(x)*(32*x^2 + 32*x - 1) + 1).

a(n) = 32*A158563(n). (End)

Maple

A158683:=n->1024*n^2-32: seq(A158683(n), n=1..50); # Wesley Ivan Hurt, Nov 20 2014

Mathematica

LinearRecurrence[{3, -3, 1}, {992, 4064, 9184}, 50] (* Vincenzo Librandi, Feb 19 2012 *)

Prog

(Magma) I:=[992, 4064, 9184]; [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 19 2012
(PARI) for(n=1, 40, print1(1024*n^2 - 32", ")); \\ Vincenzo Librandi, Feb 19 2012

Crossrefs

Cf. A005843, A158563, A158684.

Sequence in context: A172698 A109149 A237991 * A030645 A010463 A183973

Adjacent sequences: A158680 A158681 A158682 * A158684 A158685 A158686

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved