This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A214733 #58 Dec 29 2023 08:06:02
%S A214733 0,1,-1,-2,5,1,-16,13,35,-74,-31,253,-160,-599,1079,718,-3955,1801,
%T A214733 10064,-15467,-14725,61126,-16951,-166427,217280,282001,-933841,87838,
%U A214733 2713685,-2977199,-5163856,14095453,1396115,-43682474,39494129,91553293,-210035680
%N A214733 a(n) = -a(n-1) - 3*a(n-2) with n>1, a(0)=0, a(1)=1.
%C A214733 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 A214733 Apart from signs, the Lucas U(P=1,Q=3)-sequence. - _R. J. Mathar_, Oct 24 2012
%C A214733 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 A214733 Roman Witula, On Some Applications of Formulae for Unimodular Complex Numbers, Jacek Skalmierski's Press, Gliwice 2011.
%H A214733 Seiichi Manyama, <a href="/A214733/b214733.txt">Table of n, a(n) for n = 0..4189</a>
%H A214733 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 A214733 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>
%H A214733 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-3).
%F A214733 a(n) = - a(n-1) - 3*a(n-2).
%F A214733 a(n) = (-1)^n*(i*sqrt(11)/11)*(((1 + i*sqrt(11))/2)^n - ((1 - i*sqrt(11))/2)^n).
%F A214733 G.f.: x/(1 + x + 3*x^2).
%F A214733 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 A214733 From _G. C. Greubel_, Dec 28 2023: (Start)
%F A214733 a(n) = (-1)^(n-1)*3^((n-1)/2)*ChebyshevU(n-1, 1/(2*sqrt(3))).
%F A214733 a(n) = (-1)^n * A106852(n-1).
%F A214733 E.g.f.: (2/sqrt(11))*exp(-x/2)*sin(sqrt(11)*x/2). (End)
%t A214733 LinearRecurrence[{-1, -3}, {0, 1}, 40] (* _T. D. Noe_, Jul 30 2012 *)
%o A214733 (PARI) concat(0,Vec(1/(1+x+3*x^2)+O(x^99))) \\ _Charles R Greathouse IV_, Oct 01 2012
%o A214733 (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 A214733 (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 A214733 Cf. A106852, A110523.
%K A214733 sign,easy
%O A214733 0,4
%A A214733 _Roman Witula_, Jul 27 2012