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

0,2

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

Cf. A001610, A005248.

Sequence in context: A331573 A141752 A291729 * A202207 A132834 A000641

Adjacent sequences: A142583 A142584 A142585 * A142587 A142588 A142589

Keyword

nonn,easy

Author

Paul Curtz, Sep 21 2008

Extensions

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

Status

approved