A158686 a(n) = 64*n^2 + 1.

1, 65, 257, 577, 1025, 1601, 2305, 3137, 4097, 5185, 6401, 7745, 9217, 10817, 12545, 14401, 16385, 18497, 20737, 23105, 25601, 28225, 30977, 33857, 36865, 40001, 43265, 46657, 50177, 53825, 57601, 61505, 65537, 69697, 73985, 78401, 82945, 87617, 92417, 97345, 102401

Offset

0,2

Comments

The identity (64n^2+1)^2 - (1024n^2+32)*(2n)^2 = 1 can be written as a(n)^2 - A158685(n)*(A005843(n))^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..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.: -(1+62*x+65*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>=0} 1/a(n) = (coth(Pi/8)*Pi/8 + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/8)*Pi/8 + 1)/2. (End)

Mathematica

64 Range[0, 40]^2 + 1 (* or *) LinearRecurrence[{3, -3, 1}, {1, 65, 257}, 40] (* Harvey P. Dale, Jan 24 2012 *)
CoefficientList[Series[- (1 + 62 x + 65 x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)

Prog

(Magma) [64*n^2+1: n in [0..40]]; // Vincenzo Librandi, Sep 11 2013
(PARI) a(n)=64*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A005843, A158685.

Sequence in context: A237039 A305157 A038637 * A115342 A360819 A036547

Adjacent sequences: A158683 A158684 A158685 * A158687 A158688 A158689

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved