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%I #38 Dec 12 2023 08:42:18
%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). - _R. J. Mathar_, Jul 31 2010
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=6, Chi_2(n).
%D L. B. W. Jolley, Summation of Series, Dover (1961).
%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series..</a>, arXiv:1008.2547 [math.NT], 2010-2015.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,0,-1).
%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*(1+x^2) / (1 + x^2 + x^4) = x*(1+x^2) / ( (1+x+x^2)*(x^2-x+1) ).
%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
%F a(n) + a(n-2) + a(n-4) = 0 for n>3. - _Wesley Ivan Hurt_, Jun 20 2016
%F E.g.f.: 2*sin(sqrt(3)*x/2)*cosh(x/2)/sqrt(3). - _Ilya Gutkovskiy_, Jun 21 2016
%e G.f. = x - x^5 + x^7 - x^11 + x^13 - x^17 + x^19 - x^23 + x^25 - x^29 + ...
%p A134667:=n->[0, 1, 0, 0, 0, -1][(n mod 6)+1]: seq(A134667(n), n=0..100);
%p # _Wesley Ivan Hurt_, Jun 20 2016
%t a[ n_] := JacobiSymbol[-12, n]; (* _Michael Somos_, Apr 24 2014 *)
%t a[ n_] := {1, 0, 0, 0, -1, 0}[[Mod[n, 6, 1]]]; (* _Michael Somos_, Apr 24 2014 *)
%t PadRight[{},120,{0,1,0,0,0,-1}] (* _Harvey P. Dale_, Aug 01 2021 *)
%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 */
%o (Magma) &cat[[0, 1, 0, 0, 0, -1]^^20]; // _Wesley Ivan Hurt_, Jun 20 2016
%Y Cf. A093766, A120325, A131719, A131720, A131735, A131736, A214552.
%K sign,easy,mult
%O 0,1
%A _Paul Curtz_, Jan 26 2008
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