A157741 - OEIS

%I #22 Sep 08 2022 08:45:42

%S 390727,1559377,3505951,6230449,9732871,14013217,19071487,24907681,

%T 31521799,38913841,47083807,56031697,65757511,76261249,87542911,

%U 99602497,112440007,126055441,140448799,155620081,171569287,188296417

%N a(n) = 388962*n^2 + 1764*n + 1.

%C The identity (388962*n^2 + 1764*n + 1)^2 - (441*n^2 + 2*n)*(18522*n + 42)^2 = 1 can be written as a(n)^2 - A158321(n)*A157740(n)^2 = 1. - _Vincenzo Librandi_, Feb 05 2012

%C This is the case s=21 of the identity (2*n^2*s^4 + 4*n*s^2 + 1)^2 - (n^2*s^2 + 2*n)*(2*n*s^3 + 2*s)^2 = 1. - _Vincenzo Librandi_, Feb 05 2012

%H Vincenzo Librandi, <a href="/A157741/b157741.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 G.f.: x*(390727 + 387196*x + x^2)/(1-x)^3. - _Vincenzo Librandi_, Feb 05 2012

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Feb 05 2012

%F a(n) = 2*A158322(n)^2 - 1. - _Bruno Berselli_, Feb 05 2011

%t LinearRecurrence[{3, -3, 1}, {390727, 1559377, 3505951}, 50] (* _Vincenzo Librandi_, Feb 05 2012 *)

%o (Magma) I:=[390727, 1559377, 3505951]; [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_, Feb 05 2012

%o (PARI) for(n=1, 40, print1(388962*n^2 + 1764*n + 1", ")); \\ _Vincenzo Librandi_, Feb 05 2012

%Y Cf. A157740, A158321.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 05 2009