A157331 a(n) = 128*n^2 - 32*n + 1.

97, 449, 1057, 1921, 3041, 4417, 6049, 7937, 10081, 12481, 15137, 18049, 21217, 24641, 28321, 32257, 36449, 40897, 45601, 50561, 55777, 61249, 66977, 72961, 79201, 85697, 92449, 99457, 106721, 114241, 122017, 130049, 138337, 146881, 155681, 164737, 174049, 183617

Offset: 1

Author

Vincenzo Librandi, Feb 27 2009

Keywords

• nonn
• easy

Comments

The identity (128*n^2 - 32*n + 1)^2 - (4*n^2 - n)*(64*n - 8)^2 = 1 can be written as a(n)^2 - A033991(n)*A157330(n)^2 = 1 (see also the second part of the comment at A157330). - Vincenzo Librandi, Jan 29 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. A033991, A157330.

Programs

Magma

I:=[97, 449, 1057]; [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, Jan 29 2012

Mathematica

LinearRecurrence[{3,-3,1},{97,449,1057},40] (* Vincenzo Librandi, Jan 29 2012 *)

PARI

for(n=1, 40, print1(128*n^2 - 32*n + 1", ")); \\ Vincenzo Librandi, Jan 29 2012

Formula

From Vincenzo Librandi, Jan 29 2012: (Start)

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

G.f.: x*(-97 - 158*x - x^2)/(x-1)^3. (End)

E.g.f.: exp(x)*(128*x^2 + 96*x + 1) - 1. - Stefano Spezia, May 02 2024