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A144721 a(0) = 2, a(1) = 5, a(n) = 4 * a(n-1) - a(n-2).

%I A144721 #19 Sep 08 2022 08:45:38
%S A144721 2,5,18,67,250,933,3482,12995,48498,180997,675490,2520963,9408362,
%T A144721 35112485,131041578,489053827,1825173730,6811641093,25421390642,
%U A144721 94873921475,354074295258,1321423259557,4931618742970,18405051712323,68688588106322,256349300712965
%N A144721 a(0) = 2, a(1) = 5, a(n) = 4 * a(n-1) - a(n-2).
%C A144721 For all members x of the sequence, 3*x^2 - 11 is a square. - _Sture Sjöstedt_, Nov 30 2011
%H A144721 Colin Barker, <a href="/A144721/b144721.txt">Table of n, a(n) for n = 0..1000</a>
%H A144721 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).
%F A144721 Sequence satisfies -11 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v.
%F A144721 G.f.: (2 - 3*x) / (1 - 4*x + x^2). a(n) = (11 + a(n-1)^2) / a(n-2).
%F A144721 a(n) = ((2-sqrt(3))^n*(-1+2*sqrt(3))+(2+sqrt(3))^n*(1+2*sqrt(3)))/(2*sqrt(3)). - _Colin Barker_, Oct 12 2015
%t A144721 a[0] := 2; a[1] := 5; a[n_] := a[n] = 4a[n - 1] - a[n - 2]; Table[a[n], {n, 0, 24}] (* _Alonso del Arte_, Nov 30 2011 *)
%o A144721 (PARI) {a(n) = real( (2 + quadgen(12))^n * ( 2 + 1 / quadgen(12) ))}
%o A144721 (PARI) {a(n) = subst( (4*polchebyshev(n) + polchebyshev(n-1)) / 3, x, 2)}
%o A144721 (Magma) I:=[2,5]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 06 2015
%o A144721 (PARI) Vec((2-3*x)/(1-4*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 12 2015
%Y A144721 Cf. A144720(n) = a(-n).
%K A144721 nonn,easy
%O A144721 0,1
%A A144721 _Michael Somos_, Sep 19 2008