A110391 a(n) = L(3*n)/L(n), where L(n) = Lucas number.

1, 4, 6, 19, 46, 124, 321, 844, 2206, 5779, 15126, 39604, 103681, 271444, 710646, 1860499, 4870846, 12752044, 33385281, 87403804, 228826126, 599074579, 1568397606, 4106118244, 10749957121, 28143753124, 73681302246, 192900153619, 505019158606, 1322157322204

Offset: 0

Author

Amarnath Murthy, Jul 27 2005

Keywords

• easy
• nonn

Comments

Subsidiary sequences: a(n) = L((2k+1)*n)/L(n) for k = 2,3, etc. This is the sequence for k = 1.

Examples

			a(1) = L(3)/L(1) = 4/1 = 4.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A000204, A000032, A002878, A114525, A153173, A153175, A153177, A153179, A153180.

Programs

Magma

[Lucas(3*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 17 2017

Maple

with(combinat): L:=n->fibonacci(n+2)-fibonacci(n-2): seq(L(3*n)/L(n),n=0..30); # Emeric Deutsch, Jul 31 2005

Mathematica

Table[LucasL[3 n]/LucasL[n], {n, 0, 27}] (* Michael De Vlieger, Mar 18 2015 *)
LinearRecurrence[{2,2,-1},{1,4,6},40] (* Harvey P. Dale, Aug 20 2020 *)

PARI

Vec((1+2*x-4*x^2)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Jun 03 2016

PARI

for(n=0,30, print1((fibonacci(3*n+1) + fibonacci(3*n-1))/( fibonacci(n+1) + fibonacci(n-1)), ", ")) \\ G. C. Greubel, Dec 17 2017

Formula

From R. J. Mathar, Oct 18 2010: (Start)

a(n) = A005248(n) - (-1)^n.

a(n) = +2*a(n-1) +2*a(n-2) -a(n-3).

G.f.: ( 1+2*x-4*x^2 ) / ( (1+x)*(x^2-3*x+1) ). (End)

Exp( Sum_{n >= 1} a(n)*t^n/n ) = 1 + 4*t + 11*t^2 + 29*t^3 + ... is the o.g.f. for A002878. This is the case x = 1 of the general result exp( Sum_{n >= 1} L(3*n,x)/L(n,x)*t^n/n ) = Sum_{n >= 0} L(2*n + 1,x)*t^n, where L(n,x) is the n-th Lucas polynomial of A114525. - Peter Bala, Mar 18 2015

a(n) = 2^(-n)*(-(-2)^n+(3-sqrt(5))^n+(3+sqrt(5))^n). - Colin Barker, Jun 03 2016

Extensions

Corrected and extended by Emeric Deutsch and Erich Friedman, Jul 31 2005