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A083564 a(n) = L(n)*L(2n), where L(n) are the Lucas numbers (A000204). 2

%I

%S 3,21,72,329,1353,5796,24447,103729,439128,1860621,7880997,33385604,

%T 141421803,599075421,2537719272,10749959329,45537545553,192900159396,

%U 817138154247,3461452823129,14662949371128,62113250430021

%N a(n) = L(n)*L(2n), where L(n) are the Lucas numbers (A000204).

%C a(n+1)/a(n) -> (phi)^3 = ((1 + sqrt(5))/2)^3 = 4.236067...

%H Vincenzo Librandi, <a href="/A083564/b083564.txt">Table of n, a(n) for n = 1..160</a>

%H C. Pita, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pita/pita12.html">On s-Fibonomials</a>, J. Int. Seq. 14 (2011) # 11.3.7

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3, 6, -3, -1).

%F From _Benoit Cloitre_, Aug 30 2003: (Start)

%F a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4);

%F a(n) = Fibonacci(4*n)/Fibonacci(n) = A000045(4*n)/A000045(n). (End)

%F a(n) = Lucas(3*n) + (-1)^n*Lucas(n).

%F 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

%F From _R. J. Mathar_, Oct 27 2008: (Start)

%F G.f.: x*(3+12*x-9*x^2-4*x^3)/((1+x-x^2)*(1-4*x-x^2)).

%F a(n) = A061084(n+1) + 2*A001077(n). (End)

%F 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

%e a(4) = Lucas(4)*Lucas(8) = 7*47 = 329.

%t Table[Fibonacci[n*4]/Fibonacci[n],{n,50}] (* _Vladimir Joseph Stephan Orlovsky_, May 02 2011 *)

%o (MAGMA) [Lucas(n)*Lucas(2*n): n in [1..25]]; // _Vincenzo Librandi_, May 03 2011

%Y Third row of array A028412.

%K nonn,easy

%O 1,1

%A _Gary W. Adamson_, Jun 12 2003

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Last modified July 29 09:41 EDT 2021. Contains 346344 sequences. (Running on oeis4.)