A157476 a(n) = 2048n^2 + 128n + 1.

2177, 8449, 18817, 33281, 51841, 74497, 101249, 132097, 167041, 206081, 249217, 296449, 347777, 403201, 462721, 526337, 594049, 665857, 741761, 821761, 905857, 994049, 1086337, 1182721, 1283201, 1387777, 1496449, 1609217, 1726081, 1847041

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-A157474(n)*A157475(n)^2=1. [rewritten by Bruno Berselli, Aug 22 2011]

This is the case s=4 of the identity (8*n^2*s^4+8*n*s^2+1)^2 - (n^2*s^2+n)*(8*n*s^3+4*s)^2 = 1. - Bruno Berselli, Jan 25 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. A157474, A157475.

Programs

Mathematica

Table[2048n^2+128n+1,{n,30}] (* or *) LinearRecurrence[{3,-3,1},{2177,8449,18817},30] (* Harvey P. Dale, Aug 15 2011 *)

PARI

a(n)=2048*n^2+128*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

From Harvey P. Dale, Aug 15 2011: (Start)

G.f.: x*(-x^2-1918*x-2177)/(x-1)^3.

a(1)=2177, a(2)=8449, a(3)=18817, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). (End)