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A184334
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Period 6 sequence [0, 2, 2, 0, -2, -2, ...] except a(0) = 1.
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6
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1, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2, 0
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 2 * b(n) where b() is multiplicative with b(2^e) = (-1)^(e-1) if e>0, b(3^e) = 0^e, b(p^e) = 1 if p == 1 mod 6, b(p^e) = (-1)^e if p == 5 mod 6.
Euler transform of length 6 sequence [2, -1, -2, 0, 0, 1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * u * (v - 1) - (u - 1)^2 * v.
G.f.: (1 + x + x^2) / (1 - x + x^2).
a(-n) = -a(n) unless n = 0. a(n+3) = -a(n) unless n = 0 or n = -3.
G.f.: 1 / (1 - 2*x / (1 + x / (1 - x / (1 + x)))). - Michael Somos, Jan 03 2013
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EXAMPLE
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G.f. = 1 + 2*x + 2*x^2 - 2*x^4 - 2*x^5 + 2*x^7 + 2*x^8 - 2*x^10 - 2*x^11 + ...
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MATHEMATICA
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PadRight[{1}, 120, {0, 2, 2, 0, -2, -2}] (* Harvey P. Dale, Apr 02 2015 *)
a[ n_] := Boole[n == 0] + {2, 2, 0, -2, -2, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Sep 01 2015 *)
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PROG
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(PARI) {a(n) = (n==0) + [ 0, 2, 2, 0, -2, -2][n%6+1]};
(PARI) {a(n) = (n==0) + 2 * (-1)^(n\3) * sign( n%3)};
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x+x^2)/(1-x+x^2))); // G. C. Greubel, Aug 04 2018
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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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