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A014448
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Even Lucas numbers: L(3n).
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7
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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
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0,1
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COMMENTS
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This is the Lucas sequence V(4,-1). - Bruno Berselli, Jan 08 2013
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=0..22.
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).
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FORMULA
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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]
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PROG
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(PARI) polsym(x^2-4*x-1, 100)
(Sage) [lucas_number2(n, 4, -1) for n in xrange(0, 23)]# [Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(fibonacci(n+k-1)+fibonacci(n+k+1)))} [Paul D. Hanna, Oct 19 2010]
(MAGMA) [Lucas(3*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011
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CROSSREFS
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Sequence in context: A007727 A052689 A139104 * A075836 A120664 A095816
Adjacent sequences: A014445 A014446 A014447 * A014449 A014450 A014451
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KEYWORD
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nonn,easy
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AUTHOR
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Mohammad K. Azarian
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EXTENSIONS
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More terms from Erich Friedman.
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STATUS
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approved
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