A158562 a(n) = 256*n^2 - 16.

240, 1008, 2288, 4080, 6384, 9200, 12528, 16368, 20720, 25584, 30960, 36848, 43248, 50160, 57584, 65520, 73968, 82928, 92400, 102384, 112880, 123888, 135408, 147440, 159984, 173040, 186608, 200688, 215280, 230384, 246000, 262128, 278768, 295920, 313584, 331760

Offset: 1

Author

Vincenzo Librandi, Mar 21 2009

Keywords

• nonn
• easy

Comments

The identity (32*n^2 - 1)^2 - (256*n^2 - 16)*(2*n)^2 = 1 can be written as A158563(n)^2 - a(n)*A005843(n)^2 = 1. [rewritten by R. J. Mathar, Oct 16 2009]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A141759, A158563.

Programs

Magma

I:=[240,1008,2288]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

Mathematica

16(16Range[40]^2-1) (* or *) LinearRecurrence[{3,-3,1},{240,1008,2288},40] (* Harvey P. Dale, Sep 13 2011 *)

PARI

for(n=1, 50, print1(256*n^2-16", ")); \\ Vincenzo Librandi, Feb 15 2012

Formula

G.f.: 16*x*(-15 - 18*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 09 2023: (Start)

Sum_{n>=1} 1/a(n) = (4 - Pi)/128.

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)*Pi - 4)/128. (End)

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

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

a(n) = 16*A141759(n-1). (End)