

A014448


Even Lucas numbers: L(3n).


11



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

0,1


COMMENTS

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


REFERENCES

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


LINKS

Table of n, a(n) for n=0..22.
P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the kLucas Sequences and its [sic] Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 8192; sequence L_{4,n}.
Tanya Khovanova, Recursive Sequences
Wikipedia, Lucas sequence: Specific names.
Index entries for recurrences a(n) = k*a(n  1) +/ a(n  2)
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,1).


FORMULA

G.f.: (24*x)/(14*xx^2).
a(n) = 4*a(n1)+a(n2) with n>1, a(0)=2, a(1)=4.
a(n) = (2+sqrt(5))^n + (2sqrt(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]


PROG

(PARI) polsym(x^24*x1, 100)
(Sage) [lucas_number2(n, 4, 1) for n in xrange(0, 23)]# [Zerinvary Lajos, May 14 2009]
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(fibonacci(n+k1)+fibonacci(n+k+1)))} [Paul D. Hanna, Oct 19 2010]
(MAGMA) [Lucas(3*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011


CROSSREFS

Sequence in context: A226011 A052689 A139104 * A075836 A120664 A095816
Adjacent sequences: A014445 A014446 A014447 * A014449 A014450 A014451


KEYWORD

nonn,easy


AUTHOR

Mohammad K. Azarian


EXTENSIONS

More terms from Erich Friedman.


STATUS

approved



