A184334 Period 6 sequence [0, 2, 2, 0, -2, -2, ...] except a(0) = 1.

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

Offset: 0

Author

Michael Somos, Feb 13 2011

Keywords

• sign
• easy

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Rational Function Multiplicative Coefficients
• Index entries for linear recurrences with constant coefficients, signature (1,-1)

Crossrefs

Cf. A109265, A130772, A257076.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x+x^2)/(1-x+x^2))); // G. C. Greubel, Aug 04 2018

Mathematica

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 *)

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)};

Formula

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

a(n) = A130772(n-1) if n>0.

a(n) = A257076(n-1) = A109265(n-2) if n>2. - Michael Somos, Sep 01 2015