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 A106568 Expansion of 4*x/(1 - 4*x - 4*x^2). 1
 0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This sequence is part of a class of sequences with the properties: a(n) = m*(a(n-1) + a(n-2)) with a(0) = 0 and a(1) = m, g.f.: m*x/(1 - m*x - m*x^2), and have the Binet form m*(alpha^n - beta^n)/(alpha - beta) where 2*alpha = m + sqrt(m^2 + 4*m) and 2*beta = p - sqrt(m^2 + 4*m). - G. C. Greubel, Sep 06 2021 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Martin Burtscher, Igor Szczyrba and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (4,4). FORMULA a(n) = 4 * A057087(n). a(n) = A094013(n+1). - R. J. Mathar, Aug 24 2008 From Philippe Deléham, Sep 19 2009: (Start) a(n) = 4*a(n-1) + 4*a(n-2) for n > 2; a(0) = 0, a(1)=4. G.f.: 4*x/(1 - 4*x - 4*x^2). (End) G.f.: Q(0) - 1, where Q(k) = 1 + 2*(1+2*x)*x + 2*(2*k+3)*x - 2*x*(2*k+1 +2*x+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013 a(n) = 2^(n+1)*A000129(n). - G. C. Greubel, Sep 06 2021 MATHEMATICA LinearRecurrence; {4, 4}, {0, 4}, 40] (* G. C. Greubel, Sep 06 2021 *) PROG (Magma) [n le 2 select 4*(n-1) else 4*(Self(n-1) +Self(n-2)): n in [1..41]]; // G. C. Greubel, Sep 06 2021 (Sage) [2^(n+1)*lucas_number1(n, 2, -1) for n in (0..40)] # G. C. Greubel, Sep 06 2021 CROSSREFS Cf. A000129, A057087, A009013. Sequences of the form a(n) = m*(a(n-1) + a(n-2)): A000045 (m=1), A028860 (m=2), A106435 (m=3), A094013 (m=4), A106565 (m=5), A221461 (m=6), A221462 (m=7). Sequence in context: A081682 A068788 A094013 * A183146 A160564 A075581 Adjacent sequences: A106565 A106566 A106567 * A106569 A106570 A106571 KEYWORD nonn,easy,less AUTHOR Roger L. Bagula, May 30 2005 EXTENSIONS Edited by N. J. A. Sloane, Apr 30 2006 Simpler name using o.g.f. by Joerg Arndt, Oct 05 2013 STATUS approved

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Last modified December 6 16:13 EST 2023. Contains 367612 sequences. (Running on oeis4.)