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A190967 a(n) = 4*a(n-1) - 9*a(n-2), with a(0)=0, a(1)=1.

%I A190967 #46 Jul 24 2025 05:10:21
%S A190967 0,1,4,7,-8,-95,-308,-377,1264,8449,22420,13639,-147224,-711647,
%T A190967 -1521572,318535,14968288,57006337,93310756,-139814009,-1399052840,
%U A190967 -4337885279,-4760065556,20000705287,122843411152,311367297025,139878487732,-2242791722297
%N A190967 a(n) = 4*a(n-1) - 9*a(n-2), with a(0)=0, a(1)=1.
%H A190967 Michael De Vlieger, <a href="/A190967/b190967.txt">Table of n, a(n) for n = 0..2096</a>
%H A190967 Mei Bai, Wenchang Chu, and Dongwei Guo, <a href="https://doi.org/10.3390/math10152691">Reciprocal Formulae among Pell and Lucas Polynomials</a>, Mathematics, 10, 2691, (2022). See p. 5.
%H A190967 Beata Bajorska-Harapińska, Barbara Smoleń, and 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 A190967 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-9).
%F A190967 G.f.: x/(1-4*x+9*x^2). - _Philippe Deléham_, Oct 12 2011
%F A190967 a(n) = (3*i)^(n-1)*Fibonacci(n, -4*i/3), where i=sqrt(-1) and F(n,x) is the Fibonacci polynomial. - _G. C. Greubel_, Dec 07 2019
%F A190967 a(n) = (3^n*sin(n*arccos(2/3)))/sqrt(5) = 3^(n-1)*chebyshevU(n-1, 2/3). - _Federico Provvedi_, Feb 23 2022
%F A190967 E.g.f.: exp(2*x)*sin(sqrt(5)*x)/sqrt(5). - _Stefano Spezia_, Jul 24 2025
%p A190967 seq(coeff(series(x/(1-4*x+9*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Dec 07 2019
%t A190967 LinearRecurrence[{4,-9}, {0,1}, 50]
%t A190967 Table[FullSimplify[(3*I)^(n-1)*Fibonacci[n, -4*I/3]], {n,0,30}] (* _G. C. Greubel_, Dec 07 2019 *)
%o A190967 (PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-4*x+9*x^2))) \\ _G. C. Greubel_, Dec 07 2019
%o A190967 (Magma) I:=[0,1]; [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 A190967 (Sage)
%o A190967 def A190967_list(prec):
%o A190967     P.<x> = PowerSeriesRing(ZZ, prec)
%o A190967     return P( x/(1-4*x+9*x^2) ).list()
%o A190967 A190967_list(30) # _G. C. Greubel_, Dec 07 2019
%o A190967 (GAP) a:=[0,1];; 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 A190967 Cf. A190958 (index to generalized Fibonacci sequences).
%K A190967 sign,easy
%O A190967 0,3
%A A190967 _Vladimir Joseph Stephan Orlovsky_, May 24 2011