A142586 Binomial transform of A014217.

1, 2, 5, 14, 39, 107, 290, 779, 2079, 5522, 14615, 38579, 101634, 267347, 702455, 1844114, 4838079, 12686507, 33254210, 87141659, 228301839, 598026002, 1566300455, 4101923939, 10741568514, 28126975907, 73647747815, 192833044754, 504884940879, 1321888886747

Offset: 0

Author

Paul Curtz, Sep 21 2008

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001610, A005248.

Programs

Magma

[1] cat [Lucas(2*n) - 2^(n-1): n in [1..30]]; // G. C. Greubel, Apr 13 2021

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 *)

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

Sage

[1]+[lucas_number2(2*n,1,-1) -2^(n-1) for n in (1..30)] # G. C. Greubel, Apr 13 2021

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

Extensions

Edited and extended by R. J. Mathar, Sep 22 2008