A159284 Expansion of x*(1+x)/(1-x^2-2*x^3).

0, 1, 1, 1, 3, 3, 5, 9, 11, 19, 29, 41, 67, 99, 149, 233, 347, 531, 813, 1225, 1875, 2851, 4325, 6601, 10027, 15251, 23229, 35305, 53731, 81763, 124341, 189225, 287867, 437907, 666317, 1013641, 1542131, 2346275, 3569413, 5430537

Offset: 0

Author

Creighton Dement, Apr 08 2009

Keywords

• easy
• nonn

Comments

a(n) is the number of composition of n+1 into parts congruent to 0 or 2 modulo 3. - Joerg Arndt, Apr 21 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Matthias Beck and Neville Robbins, Variations on a Generatingfunctional Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence, arXiv:1403.0665 [math.NT], 2014.
• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
• Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
• Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Crossrefs

Cf. A159285, A159286, A159287, A159288.

Programs

Magma

I:=[0,1,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

Mathematica

CoefficientList[Series[x (1+x)/(1-x^2-2x^3),{x,0,50}],x] (* or *) LinearRecurrence[ {0,1,2},{0,1,1},50] (* Harvey P. Dale, Jul 16 2013 *)

PARI

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[0;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

a(n) = abs(A078028(n-1)). - R. J. Mathar, Jul 05 2012

a(n) = a(n-2) + 2*a(n-3), a(0)=0, a(1) = a(2) =1. - G. C. Greubel, Apr 30 2017

a(n) = A052947(n-1)+A052947(n-2). - R. J. Mathar, Mar 23 2023

Extensions

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021