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
KEYWORD
easy,nonn
AUTHOR
Amarnath Murthy, Jul 27 2005
EXTENSIONS
Corrected and extended by Emeric Deutsch and Erich Friedman, Jul 31 2005
STATUS
approved