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A014448 Even Lucas numbers: L(3n). 14
2, 4, 18, 76, 322, 1364, 5778, 24476, 103682, 439204, 1860498, 7881196, 33385282, 141422324, 599074578, 2537720636, 10749957122, 45537549124, 192900153618, 817138163596, 3461452808002, 14662949395604, 62113250390418 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This is the Lucas sequence V(4,-1). - Bruno Berselli, Jan 08 2013

LINKS

Table of n, a(n) for n=0..22.

P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its [sic] Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence L_{4,n}.

Tanya Khovanova, Recursive Sequences

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Wikipedia, Lucas sequence: Specific names.

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for linear recurrences with constant coefficients, signature (4,1).

FORMULA

G.f.: (2-4*x)/(1-4*x-x^2).

a(n) = 4*a(n-1)+a(n-2) with n>1, a(0)=2, a(1)=4.

a(n) = (2+sqrt(5))^n + (2-sqrt(5))^n.

a(n)/2 = A001077(n).

a(n) = A000032(3n).

a(n) = Sum_{k=0..n} C(n,k)*Lucas(n+k). - Paul D. Hanna, Oct 19 2010

a(n) = Fibonacci(6*n)/Fibonacci(3*n), n>0. - Gary Detlefs Dec 26 2010

From Peter Bala, Mar 22 2015: (Start)

a(n) = ( Fibonacci(3*n + 2*k) - F(3*n - 2*k) )/Fibonacci(2*k) for nonzero integer k.

a(n) = ( Fibonacci(3*n + 2*k + 1) + F(3*n - 2*k - 1) )/Fibonacci(2*k + 1) for arbitrary integer k. (End)

a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 20*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

a(n) = L(n)*(L(n-1)*L(n+1) + 2*(-1)^n). - J. M. Bergot, Feb 05 2016

PROG

(PARI) polsym(x^2-4*x-1, 100)

(PARI) a(n)=sum(k=0, n, binomial(n, k)*(fibonacci(n+k-1)+fibonacci(n+k+1))) \\ Paul D. Hanna, Oct 19 2010

(Sage) [lucas_number2(n, 4, -1) for n in xrange(0, 23)] # Zerinvary Lajos, May 14 2009

(MAGMA) [Lucas(3*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

CROSSREFS

Cf. A000032, A001077.

Sequence in context: A226011 A052689 A139104 * A277033 A075836 A120664

Adjacent sequences:  A014445 A014446 A014447 * A014449 A014450 A014451

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian

EXTENSIONS

More terms from Erich Friedman

STATUS

approved

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Last modified December 5 09:15 EST 2016. Contains 278762 sequences.