OFFSET
0,2
COMMENTS
The second term in the k-th iterated differences is 2, 3, 6, 10, 17, 28, 46, ... = A001610(k+1).
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (5,-7,2).
FORMULA
From R. J. Mathar, Sep 22 2008: (Start)
G.f.: (1 - 3*x + 2*x^2 + x^3)/((1-3*x+x^2)*(1-2*x)).
a(n) = A005248(n) - 2^(n-1), n>0. (End)
a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3); a(0)=1, a(1)=2, a(2)=5, a(3)=14. - Harvey P. Dale, Aug 08 2011
a(n) = (-2^(-1+n) + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n) for n > 0. - Colin Barker, Jun 05 2017
MAPLE
1, seq(combinat[fibonacci](2*n+1) +combinat[fibonacci](2*n-1) -2^(n-1), n = 1..30); # G. C. Greubel, Apr 13 2021
MATHEMATICA
CoefficientList[Series[(1-3x+2x^2+x^3)/((1-3x+x^2)(1-2x)), {x, 0, 30}], x] (* or *) Join[{1}, LinearRecurrence[{5, -7, 2}, {2, 5, 14}, 30]] (* Harvey P. Dale, Aug 08 2011 *)
PROG
(PARI) Vec((1-3*x+2*x^2+x^3)/((1-3*x+x^2)*(1-2*x)) + O(x^30)) \\ Colin Barker, Jun 05 2017
(Magma) [1] cat [Lucas(2*n) - 2^(n-1): n in [1..30]]; // G. C. Greubel, Apr 13 2021
(Sage) [1]+[lucas_number2(2*n, 1, -1) -2^(n-1) for n in (1..30)] # G. C. Greubel, Apr 13 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Sep 21 2008
EXTENSIONS
Edited and extended by R. J. Mathar, Sep 22 2008
STATUS
approved