A158684 a(n) = 64*n^2 - 1.

63, 255, 575, 1023, 1599, 2303, 3135, 4095, 5183, 6399, 7743, 9215, 10815, 12543, 14399, 16383, 18495, 20735, 23103, 25599, 28223, 30975, 33855, 36863, 39999, 43263, 46655, 50175, 53823, 57599, 61503, 65535, 69695, 73983, 78399, 82943, 87615, 92415, 97343, 102399

Offset

1,1

Comments

The identity (64*n^2 - 1)^2 - (1024*n^2 - 32)*(2*n)^2 = 1 can be written as a(n)^2 - A158683(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.: x*(-63 - 66*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/8)*Pi/8)/2 = (1 - (sqrt(2)+1)*Pi/8)/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/8)*Pi/8 - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {63, 255, 575}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
64*Range[40]^2-1 (* Harvey P. Dale, Jul 09 2017 *)

Prog

(Magma) I:=[63, 255, 575]; [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 20 2012
(PARI) for(n=1, 40, print1(64*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 20 2012

Crossrefs

Cf. A005843, A158683.

Sequence in context: A158676 A157948 A326388 * A063398 A359564 A138833

Adjacent sequences: A158681 A158682 A158683 * A158685 A158686 A158687

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved