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A129663 Denominators of the Pierce partial sums for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

%I A129663 #4 Dec 09 2016 13:20:23
%S A129663 1,1,8,26,1664,106496,370126848,7279690096640,4045738169062195200,
%T A129663 597704977138451388530688000,111845949979901797334235660288000,
%U A129663 1194765595895193218918930427630975811584000
%N A129663 Denominators of the Pierce partial sums for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
%D A129663 Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
%F A129663 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 A129663 Series: L(3, chi3) = Sum_{k=1..infinity} 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 A129663 Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
%e A129663 L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/1 - 1/(1*8) + 1/(1*8*13) - 1/(1*8*13*16) + 1/(1*8*13*16*64) - ..., the partial sums of which are 0, 1, 7/8, 23/26, 1471/1664, 94145/106496, ...
%t A129663 nmax = 100; prec = 3000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; p = First@Transpose@NestList[{Floor[ 1/(1 - #[[1]] #[[2]]) ], 1 - #[[1]] #[[2]]}&, {Floor[1/c], c}, nmax - 1]; p = Drop[ FoldList[Times, 1, p], 1 ]; Denominator[ FoldList[ Plus, 0, (-1)^Range[0, Length[p] - 1]/p ] ]
%Y A129663 Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
%Y A129663 Cf. A129658, A129659, A129660, A129661, A129662, A129664, A129665.
%K A129663 nonn,frac,easy
%O A129663 0,3
%A A129663 _Stuart Clary_, Apr 30 2007