A052922 Expansion of 1/(1 - 2*x^3 - x^4).

1, 0, 0, 2, 1, 0, 4, 4, 1, 8, 12, 6, 17, 32, 24, 40, 81, 80, 104, 202, 241, 288, 508, 684, 817, 1304, 1876, 2318, 3425, 5056, 6512, 9168, 13537, 18080, 24848, 36242, 49697, 67776, 97332, 135636, 185249, 262440, 368604, 506134, 710129, 999648, 1380872

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 907
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,1).

Crossrefs

Cf. A117413 (bijection).

Programs

GAP

a:=[1,0,0,2];; for n in [5..50] do a[n]:=2*a[n-3]+a[n-4]; od; a; # G. C. Greubel, Oct 16 2019

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1 -2*x^3 -x^4) )); // G. C. Greubel, Oct 16 2019

Maple

spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Z,Prod(Z,Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..50);
seq(coeff(series(1/(1 -2*x^3 -x^4), x, n+1), x, n), n = 0..50); # G. C. Greubel, Oct 16 2019
# alternative
A052922 := proc(n)
    if n <=3 then
        op(n+1,[1,0,0,2]) ;
    else
        2*procname(n-3)+procname(n-4) ;
    end if;
end proc:
seq(A052922(n),n=0..30) ; # R. J. Mathar, Nov 22 2024

Mathematica

LinearRecurrence[{0,0,2,1}, {1,0,0,2}, 50] (* G. C. Greubel, Oct 16 2019 *)
CoefficientList[Series[1/(1-2x^3-x^4),{x,0,50}],x] (* Harvey P. Dale, Nov 04 2024 *)

PARI

my(x='x+O('x^50)); Vec(1/(1 -2*x^3 -x^4)) \\ G. C. Greubel, Oct 16 2019

Sage

def A052922_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/(1 -2*x^3 -x^4)).list()
A052922_list(50) # G. C. Greubel, Oct 16 2019

Formula

G.f.: 1/(1 - 2*x^3 - x^4).

a(n) = 2*a(n-3) + a(n-4), with a(0)=1, a(1)=0, a(2)=0, a(3)=2.

a(n) = Sum_{alpha=RootOf(-1+2*z^3+z^4)} (1/86)*(4 +26*alpha -3*alpha^2 -6*alpha^3)*alpha^(-1-n).

Extensions

More terms from James Sellers, Jun 05 2000