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%I #8 Feb 13 2015 08:31:38
%S 1,0,-1,0,0,-1,-1,1,1,0,1,-1,0,1,1,1,1,1,0,1,1,0,-1,1,0,-1,1,-1,0,0,1,
%T -1,1,1,0,-1,1,1,0,0,1,1,1,1,-1,1,1,1,0,1,0,-1,0,0,0,1,-1,1,0,-1,0,0,
%U 0,1,0,-1,0,0,1,1,0,-1,-1,0,0,-1,0,1,0,-1,1,1,-1,1,-1,0,0,0,-1,-1,1,1,1,1,1,0,-1,1,-1,1,1,1,1,1,1,1
%N Balanced ternary expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
%C Contributed to OEIS on Apr 15 2007 --- the 300th anniversary of the birth of Leonhard Euler.
%D Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292 (for this constant); Articles 330 and 331 (for balanced ternary)
%F chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
%F Series: L(3, chi3) = sum_{k >= 1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
%F Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
%e L(3, chi3) = 0.8840238117500798567430579168710118077... = 1 + 0/3 - 1/3^2 + 0/3^3 + 0/3^4 - 1/3^5 - 1/3^6 + 1/3^7 + 1/3^8 + ...
%t nmax = 1000; prec = nmax/2 + 20 (* Normally this is sufficient precision. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Round[3(#[[2]] - #[[1]])], 3(#[[2]] - #[[1]])}&, {Round[c], c}, nmax]
%Y Cf. A129404, A129405, A129406, A129408, A129409, A129410, A129411.
%Y Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665
%K sign,easy
%O 0,1
%A _Stuart Clary_, Apr 15 2007
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