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a(n) = -a(n-1) - 3*a(n-2) with n>1, a(0)=0, a(1)=1.
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%I #58 Dec 29 2023 08:06:02

%S 0,1,-1,-2,5,1,-16,13,35,-74,-31,253,-160,-599,1079,718,-3955,1801,

%T 10064,-15467,-14725,61126,-16951,-166427,217280,282001,-933841,87838,

%U 2713685,-2977199,-5163856,14095453,1396115,-43682474,39494129,91553293,-210035680

%N a(n) = -a(n-1) - 3*a(n-2) with n>1, a(0)=0, a(1)=1.

%C The sequence a(n) is conjugate with A110523 by the following alternative relations: either ((-1 + i*sqrt(11))/2)^n = A110523(n) + a(n)*(-1 + i*sqrt(11))/2, or ((-1 - i*sqrt(11))/2)^n = A110523(n) + a(n)*(-1 - i*sqrt(11))/2 (see also comments to A110523, where these relations and many other facts on a(n) is presented).

%C Apart from signs, the Lucas U(P=1,Q=3)-sequence. - _R. J. Mathar_, Oct 24 2012

%C This is the Lucas U(-1, 3) sequence. (V_n(-1, 3))^2 + 11*(U_n(-1, 3))^2 = 4*Q^n = 4*3^n. For the special case where |U_n(-1, 3)| = 1, then, by the Lucas sequence identity U_2*n = U_n*V_n, we have (U_2*n(-1, 3))^2 + 11 = 4*3^n, true for n = 1, 2, 5, U_n = 1, -1, 1 and U_2*n = -1, 5, -31. E.g., (-31)^2 + 11 = 972 = 4*3^5. - _Raphie Frank_, Dec 09 2015

%D Roman Witula, On Some Applications of Formulae for Unimodular Complex Numbers, Jacek Skalmierski's Press, Gliwice 2011.

%H Seiichi Manyama, <a href="/A214733/b214733.txt">Table of n, a(n) for n = 0..4189</a>

%H Ronald Orozco López, <a href="https://arxiv.org/abs/2211.04450">Deformed Differential Calculus on Generalized Fibonacci Polynomials</a>, arXiv:2211.04450 [math.CO], 2022.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-3).

%F a(n) = - a(n-1) - 3*a(n-2).

%F a(n) = (-1)^n*(i*sqrt(11)/11)*(((1 + i*sqrt(11))/2)^n - ((1 - i*sqrt(11))/2)^n).

%F G.f.: x/(1 + x + 3*x^2).

%F G.f.: Q(0) -1, where Q(k) = 1 - 3*x^2 - (k+2)*x + x*(k+1 + 3*x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 07 2013

%F From _G. C. Greubel_, Dec 28 2023: (Start)

%F a(n) = (-1)^(n-1)*3^((n-1)/2)*ChebyshevU(n-1, 1/(2*sqrt(3))).

%F a(n) = (-1)^n * A106852(n-1).

%F E.g.f.: (2/sqrt(11))*exp(-x/2)*sin(sqrt(11)*x/2). (End)

%t LinearRecurrence[{-1, -3}, {0, 1}, 40] (* _T. D. Noe_, Jul 30 2012 *)

%o (PARI) concat(0,Vec(1/(1+x+3*x^2)+O(x^99))) \\ _Charles R Greathouse IV_, Oct 01 2012

%o (Magma) [n le 2 select n-1 else -Self(n-1)-3*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Dec 10 2015

%o (SageMath) [(-1)^(n-1)*round(3^((n-1)/2)*chebyshev_U(n-1, 1/(2*sqrt(3)))) for n in range(41)] # _G. C. Greubel_, Dec 28 2023

%Y Cf. A106852, A110523.

%K sign,easy

%O 0,4

%A _Roman Witula_, Jul 27 2012