%I #9 Feb 13 2015 02:24:34

%S 2,2,2,14,94,372,1391,7690,17729,49204,87816,128433,151275,290477,

%T 297212,299837,352249,897751,1081032,1646358,2402614,36591866,

%U 49132456,93538655,141789387,180474393,687775235,851204316,1868593596,7042652755

%N Engel 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

%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... = 1/2 + 1/(2*2) + 1/(2*2*2) + 1/(2*2*2*14) + 1/(2*2*2*14*94) + ...

%t nmax = 100; prec = 2000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Ceiling[1/(#[[1]] #[[2]] - 1)], #[[1]] #[[2]] - 1}&, {Ceiling[1/c], c}, nmax - 1]

%Y Cf. A129404, A129405, A129406, A129407, A129408, A129410, A129411.

%Y Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.

%K nonn,easy

%O 1,1

%A _Stuart Clary_, Apr 15 2007