A154515 - OEIS

%I #31 Sep 08 2022 08:45:40

%S 721,2737,6049,10657,16561,23761,32257,42049,53137,65521,79201,94177,

%T 110449,128017,146881,167041,188497,211249,235297,260641,287281,

%U 315217,344449,374977,406801,439921,474337,510049,547057,585361,624961,665857

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

%C 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

%H Vincenzo Librandi, <a href="/A154515/b154515.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From Colin Barker, Jan 25 2012: (Start)

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

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

%F a(n) = 2*A161705(n)^2 - 1. - _Bruno Berselli_, Jan 31 2012

%t LinearRecurrence[{3, -3, 1}, {721, 2737, 6049}, 50] (* _Vincenzo Librandi_, Jan 30 2012 *)

%o (PARI) a(n)=648*n^2+72*n+1 \\ _Charles R Greathouse IV_, Dec 27 2011

%o (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

%Y Cf. A154517, A154519.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Jan 11 2009