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 A134667 Period 6: repeat 0, 1, 0, 0, 0, -1. 6

%I

%S 0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,

%T 0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,

%U 0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-1,0,1,0

%N Period 6: repeat 0, 1, 0, 0, 0, -1.

%C Dirichlet series for the non-principal character modulo 6: L(s,chi) = sum_{n>=1} a(n)/n^s. For example L(1,chi) = A093766, L(2,chi) = A214552, and L(3,chi) = Pi^3/(18*sqrt(3)). See Jolley eq. (314) and arXiv:1008.2547 L(m=6,r=2,s). [From _R. J. Mathar_, Jul 31 2010]

%D L. B. W. Jolley, Summation of Series, Dover (1961).

%F a(n)=(1/6)*{-(n mod 6)+[(n+1) mod 6]+[(n+4) mod 6]-[(n+5) mod 6]}, with n>=0 - _Paolo P. Lava_, Jan 28 2008

%F Euler transform of length 6 sequence [ 0, 0, 0, -1, 0, 1]. - Michael Somos Feb 10 2008

%F G.f.: x * (1 - x^4) / (1 - x^6) = (x + x^3) / (1 + x^2 + x^4).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3)) where f(u, v, w) = w * (2 + v - u^2 - 2*v^2) - 2 * u * v. - Michael Somos Aug 11 2009

%F a(n) is multiplicative with a(p^e) = 0^e if p = 2 or p = 3, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6). - _Michael Somos_, Aug 11 2009

%F a(-n) = -a(n). a(n+6) = a(n). a(2*n) = a(3*n) = 0.

%F sqrt(3)*a(n) = sin(Pi*n/3) +sin(2*Pi*n/3). - _R. J. Mathar_, Oct 08 2011

%e x - x^5 + x^7 - x^11 + x^13 - x^17 + x^19 - x^23 + x^25 - x^29 + ...

%o (PARI) {a(n) = [0, 1, 0, 0, 0, -1][n%6+1]} /* Michael Somos Feb 10 2008 */

%o (PARI) {a(n) = kronecker(-12, n)} /* Michael Somos Feb 10 2008 */

%o (PARI) {a(n) = if( n < 0, -a(-n), if( n<1, 0, direuler(p=2, n, 1 / (1 - kronecker(-12, p) * X))[n]))} /* Michael Somos Aug 11 2009 */

%Y Cf. A120325, A131719, A131720, A131735, A131736.

%K sign,easy,mult

%O 0,1

%A _Paul Curtz_, Jan 26 2008

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