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A163810
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Expansion of (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) in powers of x.
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6
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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
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OFFSET
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0,1
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LINKS
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FORMULA
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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).
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EXAMPLE
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G.f. = 1 - x - x^2 + x^4 + x^5 - x^7 - x^8 + x^10 + x^11 - x^13 - x^14 + ...
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MATHEMATICA
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Join[{1}, LinearRecurrence[{1, -1}, {-1, -1}, 104]] (* Ray Chandler, Sep 15 2015 *)
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PROG
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(PARI) {a(n) = (n==0) + [0, -1, -1, 0, 1, 1][n%6 + 1]};
(PARI) {a(n) = (n==0) + (-1)^n * kronecker(-3, n)};
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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