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 A214733 a(n) = -a(n-1) - 3*a(n-2) with n>1, a(0)=0, a(1)=1. 5
 0, 1, -1, -2, 5, 1, -16, 13, 35, -74, -31, 253, -160, -599, 1079, 718, -3955, 1801, 10064, -15467, -14725, 61126, -16951, -166427, 217280, 282001, -933841, 87838, 2713685, -2977199, -5163856, 14095453, 1396115, -43682474, 39494129, 91553293, -210035680 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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). Apart from signs, the Lucas U(P=1,Q=3)-sequence. - R. J. Mathar, Oct 24 2012 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 REFERENCES R. Witula, On Some Applications of Formulae for Unimodular Complex Numbers, Jacek Skalmierski's Press, Gliwice 2011. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..4189 Wikipedia, Lucas sequence Index entries for linear recurrences with constant coefficients, signature (-1, -3). FORMULA a(n+2) = - a(n+1) - 3a(n). a(n) = (i*sqrt(11)/11)*(((-1 - i*sqrt(11))/2)^n - ((-1 + i*sqrt(11))/2)^n). G.f.: x/(1 + x + 3*x^2). 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 MATHEMATICA LinearRecurrence[{-1, -3}, {0, 1}, 40] (* T. D. Noe, Jul 30 2012 *) PROG (PARI) concat(0, Vec(1/(1+x+3*x^2)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012 (MAGMA) [n le 2 select n-1 else -Self(n-1)-3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Dec 10 2015 CROSSREFS Cf. A106852, A110523. Sequence in context: A186361 A197365 A121579 * A106852 A162975 A187244 Adjacent sequences:  A214730 A214731 A214732 * A214734 A214735 A214736 KEYWORD sign,easy AUTHOR Roman Witula, Jul 27 2012 STATUS approved

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