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A163810 Expansion of (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) in powers of x. 6
1, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
FORMULA
Euler transform of length 6 sequence [ -1, -1, -1, 0, 0, 1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * u * (1 - u) * (2 - v) - (v - u^2).
a(3*n) = 0 unless n=0. a(6*n + 1) = a(6*n + 2) = -1, a(6*n + 4) = a(6*n + 5) = a(0) = 1.
a(-n) = -a(n) unless n=0. a(n+3) = -a(n) unless n=0 or n=-3.
G.f.: (1 - x)^2 / (1 - x + x^2).
EXAMPLE
G.f. = 1 - x - x^2 + x^4 + x^5 - x^7 - x^8 + x^10 + x^11 - x^13 - x^14 + ...
MATHEMATICA
Join[{1}, LinearRecurrence[{1, -1}, {-1, -1}, 104]] (* Ray Chandler, Sep 15 2015 *)
PROG
(PARI) {a(n) = (n==0) + [0, -1, -1, 0, 1, 1][n%6 + 1]};
(PARI) {a(n) = (n==0) + (-1)^n * kronecker(-3, n)};
CROSSREFS
A163806(n) = -a(n) unless n=0. A106510(n) = (-1)^n * a(n).
Convolution inverse of A028310. Series reversion of A109081.
Sequence in context: A267208 A106510 A163806 * A163804 A181653 A343159
KEYWORD
sign,easy
AUTHOR
Michael Somos, Nov 07 2007
STATUS
approved

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Last modified July 22 22:03 EDT 2024. Contains 374544 sequences. (Running on oeis4.)