A052935 Expansion of (2-2*x-x^3)/((1-2*x)*(1-x^3)).

2, 2, 4, 9, 16, 32, 65, 128, 256, 513, 1024, 2048, 4097, 8192, 16384, 32769, 65536, 131072, 262145, 524288, 1048576, 2097153, 4194304, 8388608, 16777217, 33554432, 67108864, 134217729, 268435456, 536870912, 1073741825, 2147483648

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 923
• Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

Programs

GAP

a:=[2,2,4,9];; for n in [5..40] do a[n]:=2*a[n-1]+a[n-3]-2*a[n-4]; od; a; # G. C. Greubel, Oct 18 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-2*x-x^3)/((1-2*x)*(1-x^3)) )); // G. C. Greubel, Oct 18 2019

Maple

spec:= [S,{S=Union(Sequence(Prod(Z,Z,Z)),Sequence(Union(Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((2-2*x-x^3)/((1-2*x)*(1-x^3)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 18 2019

Mathematica

CoefficientList[Series[(2-2*x-x^3)/((1-2*x)*(1-x^3)), {x, 0, 40}], x] (* G. C. Greubel, Oct 05 2017 *)
LinearRecurrence[{2,0,1,-2}, {2,2,4,9}, 40] (* G. C. Greubel, Oct 18 2019 *)

PARI

my(x='x+O('x^40)); Vec((2-2*x-x^3)/((1-2*x)*(1-x^3))) \\ G. C. Greubel, Oct 05 2017

Sage

def A052935_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((2-2*x-x^3)/((1-2*x)*(1-x^3))).list()
A052935_list(40) # G. C. Greubel, Oct 18 2019

Formula

G.f.: (2-2*x-x^3)/((1-x^3)*(1-2*x)).

a(n) = a(n-1) + a(n-2) + 2*a(n-3) - 1.

a(n) = 2^n + Sum_{alpha=RootOf(-1+z^3)} alpha^(-n)/3.

a(n) = 2*A033138(n+1) - 2*A033138(n) - A033138(n-2). - R. J. Mathar, Nov 28 2011

Extensions

More terms from James Sellers, Jun 05 2000