A123891 Expansion of (1-3*x^2+x^3)/(1-3*x+x^3).

1, 3, 6, 18, 51, 147, 423, 1218, 3507, 10098, 29076, 83721, 241065, 694119, 1998636, 5754843, 16570410, 47712594, 137382939, 395578407, 1139022627, 3279684942, 9443476419, 27191406630, 78294534948, 225440128425, 649128978645, 1869092400987, 5381837074536

Offset: 0

Author

N. J. A. Sloane, Nov 20 2006

Keywords

• nonn

References

• A. Burstein and T. Mansour, Words restricted by 3-letter ..., Annals. Combin., 7 (2003), 1-14.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Crossrefs

Cf. A052536.

Programs

GAP

a:=[3,6,18];; for n in [4..30] do a[n]:=3*a[n-1]-a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Aug 07 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-3*x^2+x^3)/(1-3*x+x^3) )); // G. C. Greubel, Aug 07 2019

Maple

seq(coeff(series((1-3*x^2+x^3)/(1-3*x+x^3), x, n+1), x, n), n = 0..40); # G. C. Greubel, Aug 07 2019

Mathematica

Join[{1}, LinearRecurrence[{3, 0, -1}, {3, 6, 18}, 28]] (* Jean-François Alcover, Oct 08 2018 *)
CoefficientList[Series[(1-3x^2+x^3)/(1-3x+x^3),{x,0,40}],x] (* Harvey P. Dale, Jan 16 2022 *)

PARI

my(x='x+O('x^30)); Vec((1-3*x^2+x^3)/(1-3*x+x^3)) \\ G. C. Greubel, Aug 07 2019

Sage

((1-3*x^2+x^3)/(1-3*x+x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 07 2019

Formula

a(n) = 3*A052536(n-1), n>0. - R. J. Mathar, Sep 27 2014