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A083564
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a(n) = L(n)*L(2n), where L(n) are the Lucas numbers (A000204).
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2
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3, 21, 72, 329, 1353, 5796, 24447, 103729, 439128, 1860621, 7880997, 33385604, 141421803, 599075421, 2537719272, 10749959329, 45537545553, 192900159396, 817138154247, 3461452823129, 14662949371128, 62113250430021
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OFFSET
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1,1
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COMMENTS
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a(n+1)/a(n) -> (phi)^3 = ((1 + sqrt(5))/2)^3 = 4.236067...
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..160
C. Pita, On s-Fibonomials, J. Int. Seq. 14 (2011) # 11.3.7
Index entries for linear recurrences with constant coefficients, signature (3, 6, -3, -1).
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FORMULA
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From Benoit Cloitre, Aug 30 2003: (Start)
a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4);
a(n) = Fibonacci(4*n)/Fibonacci(n) = A000045(4*n)/A000045(n). (End)
a(n) = Lucas(3*n) + (-1)^n*Lucas(n).
a(n) = 2*(2-sqrt(5))^n - (1/2)*(-1/2-(1/2)*sqrt(5))^n - (1/2)*(-1/2-(1/2)*sqrt(5))^n*sqrt(5) + 2*(2+sqrt(5))^n + (1/2)*sqrt(5)*(-1/2+(1/2)*sqrt(5))^n - (1/2)*(-1/2+(1/2)*sqrt(5))^n + (2+sqrt(5))^n*sqrt(5) - (2-sqrt(5))^n*sqrt(5), with n>=0. - Paolo P. Lava, Jun 12 2008
From R. J. Mathar, Oct 27 2008: (Start)
G.f.: x*(3+12*x-9*x^2-4*x^3)/((1+x-x^2)*(1-4*x-x^2)).
a(n) = A061084(n+1) + 2*A001077(n). (End)
a(n) = (1+phi)^n + (-phi)^n + (2*phi+1)^n + (3-2*phi)^n, phi = (1+sqrt(5))/2. - Gary Detlefs, Dec 09 2012
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EXAMPLE
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a(4) = Lucas(4)*Lucas(8) = 7*47 = 329.
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MATHEMATICA
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Table[Fibonacci[n*4]/Fibonacci[n], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)
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PROG
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(MAGMA) [Lucas(n)*Lucas(2*n): n in [1..25]]; // Vincenzo Librandi, May 03 2011
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CROSSREFS
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Third row of array A028412.
Sequence in context: A145658 A342548 A188667 * A281008 A238193 A054646
Adjacent sequences: A083561 A083562 A083563 * A083565 A083566 A083567
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KEYWORD
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nonn,easy
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AUTHOR
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Gary W. Adamson, Jun 12 2003
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STATUS
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approved
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