A157448 a(n) = 2048*n^2 - 128*n + 1.

1921, 7937, 18049, 32257, 50561, 72961, 99457, 130049, 164737, 203521, 246401, 293377, 344449, 399617, 458881, 522241, 589697, 661249, 736897, 816641, 900481, 988417, 1080449, 1176577, 1276801, 1381121, 1489537, 1602049, 1718657, 1839361

Offset: 1

Author

Vincenzo Librandi, Mar 01 2009

Keywords

• nonn
• easy

Comments

The identity (2048*n^2 - 128*n + 1)^2 - (16*n^2 - n)*(512*n - 16)^2 = 1 can be written as a(n)^2 - A157446(n)*A157447(n)^2 = 1 (see also second comment at A157446). - Vincenzo Librandi, Jan 26 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Vincenzo Librandi, X^2-AY^2=1
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A157446, A157447.

Programs

Magma

I:=[1921, 7937, 18049]; [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, Jan 26 2012

Mathematica

LinearRecurrence[{3,-3,1},{1921,7937,18049},40] (* Vincenzo Librandi, Jan 26 2012 *)

PARI

for(n=1, 22, print1(2048*n^2 - 128*n + 1", ")); \\ Vincenzo Librandi, Jan 26 2012

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 26 2012

G.f.: x*(-1921 - 2174*x - x^2)/(x-1)^3. - Vincenzo Librandi, Jan 26 2012