A154515 a(n) = 648*n^2 + 72*n + 1.

721, 2737, 6049, 10657, 16561, 23761, 32257, 42049, 53137, 65521, 79201, 94177, 110449, 128017, 146881, 167041, 188497, 211249, 235297, 260641, 287281, 315217, 344449, 374977, 406801, 439921, 474337, 510049, 547057, 585361, 624961, 665857, 708049, 751537

Offset: 1

Author

Vincenzo Librandi, Jan 11 2009

Keywords

• nonn
• easy

Comments

The identity (648*n^2 + 72*n + 1)^2 - (9*n^2 + n)*(216*n + 12)^2 = 1 can be written as a(n)^2 - A154517(n)*A154519(n)^2 = 1. This is the case s=3 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. - Vincenzo Librandi, Jan 30 2012

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. A154517, A154519.

Programs

Magma

I:=[721, 2737, 6049]; [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 30 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {721, 2737, 6049}, 50] (* Vincenzo Librandi, Jan 30 2012 *)

PARI

a(n)=648*n^2+72*n+1 \\ Charles R Greathouse IV, Dec 27 2011

Formula

From Colin Barker, Jan 25 2012: (Start)

G.f.: x*(721 + 574*x + x^2)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=721, a(2)=2737, a(3)=6049. (End)

a(n) = 2*A161705(n)^2 - 1. - Bruno Berselli, Jan 31 2012