A124395 Expansion of (1-2*x)/(1-2*x+2*x^3).

1, 0, 0, -2, -4, -8, -12, -16, -16, -8, 16, 64, 144, 256, 384, 480, 448, 128, -704, -2304, -4864, -8320, -12032, -14336, -12032, 0, 28672, 81408, 162816, 268288, 373760, 421888, 307200, -133120, -1110016, -2834432, -5402624, -8585216, -11501568, -12197888

Offset: 0

Author

Paul Barry, Oct 30 2006

Keywords

• easy
• sign

Comments

Diagonal sums of number array A124394.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2).

Crossrefs

Cf. A077940.

Programs

GAP

a:=[1,0,0];; for n in [3..50] do a[n]:=2*a[n-1]-2*a[n-3]; od; a; # G. C. Greubel, Dec 25 2019

Magma

I:=[1,0,0]; [n le 3 select I[n] else 2*Self(n-1) - 2*Self(n-3): n in [1..50]]; // G. C. Greubel, Dec 25 2019

Maple

a:= n-> (Matrix([[2, 1, 0], [0, 0, 1], [-2, 0, 0]])^n)[2, 2]: seq (a(n), n=0..35);  # Alois P. Heinz, Sep 10 2008

Mathematica

CoefficientList[Series[(1-2x)/(1-2x+2x^3),{x,0,50}],x] (* or *) LinearRecurrence[{2,0,-2},{1,0,0},50] (* Harvey P. Dale, Dec 21 2013 *)

PARI

my(x='x+O('x^50)); Vec((1-2*x)/(1-2*x+2*x^3)) \\ G. C. Greubel, Dec 25 2019

Sage

def A124395_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x)/(1-2*x+2*x^3) ).list()
A124395_list(50) # G. C. Greubel, Dec 25 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum{j=0..k+1} C(k+1,j)*C(n-j+1,2k+1)*(-2)^j.

a(n) = term (2,2) in the 3 X 3 matrix [2,1,0; 0,0,1; -2,0,0]^n. - Alois P. Heinz, Sep 10 2008

a(n) = 2*a(n-1) - 2*a(n-3); a(0)=1, a(1)=0, a(2)=0. - Harvey P. Dale, Dec 21 2013

a(n) = A077940(n) - 2*A077940(n-1). - R. J. Mathar, Jan 25 2016