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A014448
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Even Lucas numbers: L(3n).
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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REFERENCES
| 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
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
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FORMULA
| G.f.: (2-4*x)/(1-4*x-x^2); a(n)=4*a(n-1)+a(n-2), a(0)=2, a(1)=4; a(n)=(2+sqrt(5))^n + (2-sqrt(5))^n.
a(n) = Sum_{k=0..n} C(n,k)*Lucas(n+k). [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 19 2010]
a(n)=Fibonacci(6*n)/Fibonacci(3*n), n>0,[From Gary Detlefs (gdetlefs(AT)aol.com) Dec 26 2010]
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PROG
| (PARI) polsym(x^2-4*x-1, 100)
(Other) sage: [lucas_number2(n, 4, -1) for n in xrange(0, 23)]# [From 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)))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 19 2010]
(MAGMA) [ Lucas(3*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011
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CROSSREFS
| A001077(n)=A014448(n)/2. A014448(n)=A000032(3n).
Sequence in context: A007727 A052689 A139104 * A075836 A120664 A095816
Adjacent sequences: A014445 A014446 A014447 * A014449 A014450 A014451
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KEYWORD
| nonn,easy
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AUTHOR
| Mohammad K. Azarian (ma3(AT)evansville.edu)
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EXTENSIONS
| More terms from Erich Friedman (erich.friedman(AT)stetson.edu).
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