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A327767
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Period 2: repeat [1, -2].
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3
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1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f.: x * (1 - 2*x) / (1 - x^2) = x / (1 + 2*x / (1 - 3*x / (2 - x))).
E.g.f.: (exp(x) - 1)*(3/exp(x) - 1)/2.
a(n) is multiplicative with a(2^e) = -2 if e>0, a(p^e) = 1 otherwise.
Moebius transform is length 2 sequence [1, -3].
a(n) = -(1 + 3*(-1)^n)/2 if n>=1.
a(2*n) = -2, a(2*n + 1) = 1, a(0) = 0.
a(n) = -(-1)^n * A134451(n) for all n in Z.
Dirichlet g.f.: zeta(s)*(1-3/2^s). - Amiram Eldar, Jan 03 2023
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EXAMPLE
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G.f. = x - 2*x^2 + x^3 - 2*x^4 + x^5 - 2*x^6 + x^7 - 2*x^8 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, -2 + 3 Mod[n, 2]];
a[ n_] := Which[ n < 1, 0, OddQ[n], 1, True, -2];
a[ n_] := SeriesCoefficient[ (x - 2*x^2) / (1 - x^2), {x, 0, n}];
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PROG
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(PARI) {a(n) = if( n<1, 0, -(1 + 3*(-1)^n)/2)};
(PARI) {a(n) = if( n<1, 0, -2 + 3*(n%2))};
(PARI) {a(n) = if( n<1, 0, [-2, 1][n%2 + 1])};
(PARI) {a(n) = if( n<0, 0, polcoeff( (x - 2*x^2) / (1 - x^2) + x * O(x^n), n))};
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CROSSREFS
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KEYWORD
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sign,mult,easy
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AUTHOR
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STATUS
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approved
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