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

0,2

Comments

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

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

Example

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

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 *)

Prog

(PARI) Vec((1+2*x-4*x^2)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Jun 03 2016
(Magma) [Lucas(3*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 17 2017
(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

Crossrefs

Cf. A000204, A000032, A002878, A144525, A153173, A153175, A153177, A153179, A153180.

Sequence in context: A006534 A064035 A010364 * A197460 A001683 A053892

Adjacent sequences: A110388 A110389 A110390 * A110392 A110393 A110394

Keyword

easy,nonn

Author

Amarnath Murthy, Jul 27 2005

Extensions

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

Status

approved