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A112690
Expansion of 1/(1 + x^2 - x^3 - x^5).
1
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
0,1
COMMENTS
Partial sums are A112689.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Sep 15 2005
STATUS
approved