0,2

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Gaurav Bhatnagar, Analogues of a Fibonacci-Lucas Identity, The Fibonacci Quarterly, Vol. 54, No. 2 (2016), pp. 166-171.

Tanya Khovanova, Recursive Sequences.

Igor Szczyrba, Rafał Szczyrba, and Martin Burtscher, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

Index entries for linear recurrences with constant coefficients, signature (4,4).

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

a(n) = 4^n*hypergeom([(1-n)/2, 1-n/2], [1-n], -1) for n > 0. - Peter Luschny, Mar 30 2025

a(n) = Sum_{k=0..n-1} 2^k * (A002203(k) + 2*A000129(k+1)) (Bhatnagar, 2016, p. 169, eq. (3.1)). - Amiram Eldar, Jan 10 2026

A106568 := n -> ifelse(n=0, 0, 4^(n)*hypergeom([(1-n)/2, 1-n/2], [1-n], -1)):

seq(simplify(A106568(n)), n = 0..24); # Peter Luschny, Mar 30 2025

LinearRecurrence[{4, 4}, {0, 4}, 40] (* G. C. Greubel, Sep 06 2021 *)

(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

(SageMath) [2^(n+1)*lucas_number1(n, 2, -1) for n in (0..40)] # G. C. Greubel, Sep 06 2021

Cf. A000129, A002203, 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: A068788 A390612 A094013 * A183146 A160564 A075581

Adjacent sequences: A106565 A106566 A106567 * A106569 A106570 A106571

nonn,easy,less

Roger L. Bagula, May 30 2005

Edited by N. J. A. Sloane, Apr 30 2006

Simpler name using o.g.f. by Joerg Arndt, Oct 05 2013

approved