A106568 Expansion of 4*x/(1 - 4*x - 4*x^2).

0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360, 782005818097664, 3775858103812096, 18231455687639040

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

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).

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

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

Maple

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

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

Crossrefs

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

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