A158685 a(n) = 32*(32*n^2 + 1).

32, 1056, 4128, 9248, 16416, 25632, 36896, 50208, 65568, 82976, 102432, 123936, 147488, 173088, 200736, 230432, 262176, 295968, 331808, 369696, 409632, 451616, 495648, 541728, 589856, 640032, 692256, 746528, 802848, 861216, 921632, 984096, 1048608, 1115168, 1183776

Offset: 0

Author

Vincenzo Librandi, Mar 24 2009

Keywords

• nonn
• easy

Comments

The identity (64*n^2 + 1)^2 - (1024*n^2 + 32)*(2*n)^2 = 1 can be written as A158686(n)^2 - a(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).

Crossrefs

Cf. A005843, A158575, A158686.

Programs

Magma

[32*(32*n^2+1): n in [0..40]]; // Vincenzo Librandi, Sep 11 2013

Mathematica

CoefficientList[Series[ - 32 (1 + 30 x + 33 x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)

PARI

a(n)=32*(32*n^2+1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: -32*(1+30*x+33*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/(4*sqrt(2)))*Pi/(4*sqrt(2)) + 1)/64.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) + 1)/64. (End)

From Elmo R. Oliveira, Jan 15 2025: (Start)

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

a(n) = 32*A158575(n). (End)

Extensions

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