A157912 a(n) = 64*n^2 + 16.

80, 272, 592, 1040, 1616, 2320, 3152, 4112, 5200, 6416, 7760, 9232, 10832, 12560, 14416, 16400, 18512, 20752, 23120, 25616, 28240, 30992, 33872, 36880, 40016, 43280, 46672, 50192, 53840, 57616, 61520, 65552, 69712, 74000, 78416, 82960, 87632, 92432, 97360, 102416

Offset: 1

Author

Vincenzo Librandi, Mar 09 2009

Keywords

• nonn
• easy

Comments

The identity (8*n^2 + 1)^2 - (64*n^2 + 16)*n^2 = 1 can be written as A081585(n)^2 - a(n)*n^2 = 1. - Vincenzo Librandi, Feb 09 2012

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).

Crossrefs

Cf. A053755, A081585.

Programs

Magma

I:=[80, 272, 592]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 09 2012

Maple

A157912:=n->64*n^2 + 16: seq(A157912(n), n=1..50); # Wesley Ivan Hurt, Oct 27 2014

Mathematica

64Range[50]^2+16  (* Harvey P. Dale, Mar 24 2011 *)
LinearRecurrence[{3, -3, 1}, {80, 272, 592}, 40] (* Vincenzo Librandi, Feb 09 2012 *)

PARI

for(n=1, 40, print1(64*n^2 + 16", ")); \\ Vincenzo Librandi, Feb 09 2012

Formula

From Vincenzo Librandi, Feb 09 2012: (Start)

G.f.: x*(80+32*x+16*x^2)/(1-x)^3.

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

From Amiram Eldar, Mar 07 2023: (Start)

Sum_{n>=1} 1/a(n) = (coth(Pi/2)*Pi/2 - 1)/32.

Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/2)*Pi/2)/32. (End)

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

E.g.f.: 16*(exp(x)*(4*x^2 + 4*x + 1) - 1).

a(n) = 16*A053755(n). (End)