A112690 Expansion of 1/(1 + x^2 - x^3 - x^5).

0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1

Offset: 0

Author

Paul Barry, Sep 15 2005

Keywords

• easy
• sign

Comments

Partial sums are A112689.

Links

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

Crossrefs

Cf. A000100, A057077, A061347, A112689.

Programs

Mathematica

LinearRecurrence[{0, -1, 1, 0, 1}, {0, 1, 0, -1, 1}, 100] (* Vincenzo Librandi, Jul 07 2016 *)
Join[{0},CoefficientList[Series[1/(1+x^2-x^3-x^5),{x,0,100}],x]] (* Harvey P. Dale, Jul 30 2024 *)

PARI

concat(0, Vec(1/(1+x^2-x^3-x^5) + O(x^80))) \\ Michel Marcus, Jul 07 2016

PARI

a(n) = round(real((exp(-2/3*I*n*Pi)*(-4+(3+3*I)*exp((I*n*Pi)/6) + 2*exp((2*I*n*Pi)/3) + (3-3*I)*exp((7*I*n*Pi)/6) - 4*exp((4*I*n*Pi)/3)))/12)) \\ Colin Barker, Jul 07 2016

Formula

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

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

a(n+12) = a(n) = 1/6 + A057077(n+1)/2 + A061347(n+1)/3. - R. J. Mathar, Feb 23 2009

a(n+10) = (A000100(n) mod 2)*(-1)^(1 + floor(n/2)). - John M. Campbell, Jul 07 2016

From Ilya Gutkovskiy, Jul 07 2016: (Start)

E.g.f.: (3*sin(x) + 3*cos(x) + exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/6.

a(n) = (3*sin(Pi*n/2) + 3*cos(Pi*n/2) - 4*cos(2*Pi*n/3) + 1)/6. (End)

a(n) = 2*floor(n/4) + floor((n+2)/3) - floor(n/3) - floor(n/2). - Ridouane Oudra, Mar 11 2023