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.

0, 1, 3, 7, 99, 9307, 3462205, 401327263, 5290639975663, 21886143096656843, 32306573547837099089161, 2837034062676862693613762377, 182184397885888753164448171682621

Offset

0,3

References

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292

Links

Table of n, a(n) for n=0..12.

Formula

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.

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 + ...

Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).

Example

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, ...

Mathematica

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 ] ] ]

Crossrefs

Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.

Cf. A129658, A129659, A129661, A129662, A129663, A129664, A129665.

Sequence in context: A159310 A362682 A299377 * A243734 A372483 A158467

Adjacent sequences: A129657 A129658 A129659 * A129661 A129662 A129663

Keyword

nonn,frac,easy

Author

Stuart Clary, Apr 30 2007

Status

approved