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%I #50 Oct 11 2022 00:59:41
%S 3,9,33,123,459,1713,6393,23859,89043,332313,1240209,4628523,17273883,
%T 64467009,240594153,897909603,3351044259,12506267433,46674025473,
%U 174189834459,650085312363,2426151414993,9054520347609,33791929975443
%N a(n) = 4*a(n-1) - a(n-2) for n>1, a(0)=3, a(1)=9.
%C y-values in the solutions to 3*x^2+6 = y^2. - _Sture Sjöstedt_, Nov 25 2011
%C Positive values of x (or y) satisfying x^2 - 4*x*y + y^2 + 18 = 0. - _Colin Barker_, Feb 04 2014
%C Positive values of x (or y) satisfying x^2 - 14*x*y + y^2 + 288 = 0. - _Colin Barker_, Feb 16 2014
%H Vincenzo Librandi, <a href="/A082841/b082841.txt">Table of n, a(n) for n = 0..200</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).
%F G.f.: (3 -6*x +3*x^2)/((1-x)*(1-4*x+x^2)).
%F a(n) = sqrt(3/2)*(a^(n+1/2) + b^(n+1/2)), with a=2+sqrt(3) and b=2-sqrt(3).
%F a(n) = sqrt(3*(11 +12*A082840(n) +4*A082840(n)^2)).
%F a(n) = sqrt((3/2)*(A003500(2n+1) +2)).
%F a(n) - a(n-1) = 6*A001353(n).
%F a(n) == 3 (mod 6).
%F a(n) = 3 * A001835(n+1).
%F a(n) = 3*x(n) + 3*y(n) for x(n)= A001075(n) and y(n) = A001353(n) the solutions to x^2 - 3*y^2 = 1. - _Greg Dresden_ and his Math 222 Linear Algebra class, Oct 05 2022
%p a:=proc(n) option remember; if n=0 then 3 elif n=1 then 9 else 4*a(n-1)-a(n-2); fi; end: seq(a(n), n=0..40); # _Wesley Ivan Hurt_, Jan 21 2017
%t CoefficientList[Series[(3-6x+3x^2)/((1-x)(1-4x+x^2)), {x, 0, 25}], x]
%t LinearRecurrence[{4,-1},{3,9},30] (* _Harvey P. Dale_, Aug 28 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((3-6*x+3*x^2)/((1-x)*(1-4*x+x^2))) \\ _G. C. Greubel_, Feb 25 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (3-6*x+3*x^2)/((1-x)*(1-4*x+x^2)) )); // _G. C. Greubel_, Feb 25 2019
%o (Sage) ((3-6*x+3*x^2)/((1-x)*(1-4*x+x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 25 2019
%o (GAP) a:=[3,9];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Feb 25 2019
%Y First differences of A005320.
%Y Cf. A001834.
%K nonn,easy
%O 0,1
%A Mario Catalani (mario.catalani(AT)unito.it), Apr 14 2003
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