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

3, 21, 72, 329, 1353, 5796, 24447, 103729, 439128, 1860621, 7880997, 33385604, 141421803, 599075421, 2537719272, 10749959329, 45537545553, 192900159396, 817138154247, 3461452823129, 14662949371128, 62113250430021

Offset: 1

Author

Gary W. Adamson, Jun 12 2003

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

• 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).

Crossrefs

Third row of array A028412.

Programs

Magma

[Lucas(n)*Lucas(2*n): n in [1..25]]; // Vincenzo Librandi, May 03 2011

Mathematica

Table[Fibonacci[n*4]/Fibonacci[n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)

Formula

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).

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