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A176333 Expansion of (1-3*x)/(1-4*x+9*x^2).

%I A176333 #25 Oct 13 2024 17:42:11
%S A176333 1,1,-5,-29,-71,-23,547,2395,4657,-2927,-53621,-188141,-269975,613369,
%T A176333 4883251,14012683,12101473,-77708255,-419746277,-979610813,-140726759,
%U A176333 8253590281,34280901955,62841295291,-57162936431,-794223403343,-2662427185493,-3501698111885
%N A176333 Expansion of (1-3*x)/(1-4*x+9*x^2).
%C A176333 Hankel transform of A176332.
%H A176333 Michael De Vlieger, <a href="/A176333/b176333.txt">Table of n, a(n) for n = 0..2097</a>
%H A176333 Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, <a href="https://doi.org/10.1007/s00006-019-0969-9">On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis</a>, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
%H A176333 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-9).
%F A176333 a(n) = 3^n*( cos(2*n*atan(1/sqrt(5))) - sin(2*n*atan(1/sqrt(5)))/sqrt(5) ).
%F A176333 a(0)=1, a(1)=1, a(n) = 4*a(n-1) - 9*a(n-2). - _Harvey P. Dale_, Sep 17 2012
%F A176333 a(n) = -3*A190967(n) + A190967(n+1). - _R. J. Mathar_, May 04 2013
%p A176333 seq(coeff(series((1-3*x)/(1-4*x+9*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Dec 07 2019
%t A176333 CoefficientList[Series[(1-3x)/(1-4x+9x^2),{x,0,30}],x] (* or *) LinearRecurrence[{4,-9},{1,1},30] (* _Harvey P. Dale_, Sep 17 2012 *)
%o A176333 (PARI) my(x='x+O('x^30)); Vec((1-3*x)/(1-4*x+9*x^2)) \\ _G. C. Greubel_, Dec 07 2019
%o A176333 (Magma) I:=[1,1]; [n le 2 select I[n] else 4*Self(n-1) - 9*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 07 2019
%o A176333 (Sage)
%o A176333 def A176333_list(prec):
%o A176333     P.<x> = PowerSeriesRing(ZZ, prec)
%o A176333     return P( (1-3*x)/(1-4*x+9*x^2) ).list()
%o A176333 A176333_list(30) # _G. C. Greubel_, Dec 07 2019
%o A176333 (GAP) a:=[1,1];; for n in [3..30] do a[n]:=4*a[n-1]-9*a[n-2]; od; a; # _G. C. Greubel_, Dec 07 2019
%Y A176333 Cf. A176332, A190958, A190967.
%K A176333 easy,sign
%O A176333 0,3
%A A176333 _Paul Barry_, Apr 15 2010