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
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