A014448 Even Lucas numbers: L(3n).

2, 4, 18, 76, 322, 1364, 5778, 24476, 103682, 439204, 1860498, 7881196, 33385282, 141422324, 599074578, 2537720636, 10749957122, 45537549124, 192900153618, 817138163596, 3461452808002, 14662949395604, 62113250390418

Offset

0,1

Comments

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

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1591

Pooja Bhadouria, Deepika Jhala and Bijendra Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1 (2014), pp. 81-92; sequence L_{4,n}.

H. H. Ferns, Problem B-115, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 2 (1967), p. 202; Identities for F_{kn} and L{kn}, Solution to Problem B-115 by Stanley Rabinowitz, ibid., Vol. 6, No. 1 (1968), pp. 92-93.

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(3*n).

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

From Peter Bala Oct 14 2019: (Start)

Sum_{n >= 1} 1/( a(n) + (-1)^(n+1)*20/a(n) ) = 3/16.

Sum_{n >= 1} (-1)^(n+1)/( a(n) + (-1)^(n+1)*20/a(n) ) = 1/16. (End)

a(n) = (15*Fibonacci(n)^2*Lucas(n) + Lucas(n)^3)/4 (Ferns, 1967). - Amiram Eldar, Feb 06 2022

Example

a(4) = L(3 * 4) = L(12) = 322. - Indranil Ghosh, Feb 05 2017

Mathematica

Table[LucasL[3*n], {n, 0, 100}] (* G. C. Greubel, Nov 07 2018 *)

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 range(0, 23)] # Zerinvary Lajos, May 14 2009
(Magma) [Lucas(3*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

Crossrefs

Cf. A000032, A000045, A001077.

Cf. Lucas(k*n): A005248 (k = 2), A056854 (k = 4), A001946 (k = 5), A087215 (k = 6), A087281 (k = 7), A087265 (k = 8), A087287 (k = 9), A089772 (k = 11), A089775 (k = 12).

Sequence in context: A325850 A052689 A139104 * A277033 A295767 A318230

Adjacent sequences: A014445 A014446 A014447 * A014449 A014450 A014451

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from Erich Friedman

Status

approved