%I A129660
%S A129660 0,1,3,7,99,9307,3462205,401327263,5290639975663,21886143096656843,
%T A129660 32306573547837099089161,2837034062676862693613762377,
%U A129660 182184397885888753164448171682621
%N A129660 Numerators of the Engel partial sums for L(3, chi3), where L(s, chi3) 
               is the Dirichlet L-function for the non-principal character modulo 
               3.
%D A129660 Leonhard Euler, ``Introductio in Analysin Infinitorum'', First Part, 
               Articles 176 and 292
%F A129660 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 A129660 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 A129660 Closed form: L(3, chi3) = 4 pi^3/(81 sqrt(3))
%e A129660 L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/2 + 1/(2*2) 
               + 1/(2*2*2) + 1/(2*2*2*14) + 1/(2*2*2*14*94) + ..., the partial sums 
               of which are 0, 1/2, 3/4, 7/8, 99/112, 9307/10528, ...
%t A129660 nmax = 100; prec = 2000 (* Adjust the precision depending on nmax. *); 
               c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; e = First@Transpose@NestList[{Ceiling[1/
               (#[[1]] #[[2]] - 1)], #[[1]] #[[2]] - 1}&, {Ceiling[1/c], c}, nmax 
               - 1]; Numerator[ FoldList[Plus, 0, 1/Drop[ FoldList[Times, 1, e], 
               1 ] ] ]
%Y A129660 Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
%Y A129660 Cf. A129658, A129659, A129661, A129662, A129663, A129664, A129665.
%Y A129660 Sequence in context: A062592 A074349 A159310 this_sequence A158467 A028414 
               A014014
%Y A129660 Adjacent sequences: A129657 A129658 A129659 this_sequence A129661 A129662 
               A129663
%K A129660 nonn,frac,easy
%O A129660 0,3
%A A129660 Stuart Clary (clary(AT)uakron.edu), Apr 30, 2007