A129410 Pierce expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

1, 8, 13, 16, 64, 6951, 206515, 344040, 11364380, 14595803, 136951831, 417525297, 691111129, 982473113, 15154864245, 17661539909, 31435459113, 49634203300, 1454188399688, 2112564552862, 2266989878695, 5056833185437, 8740145960744

Offset

1,2

Comments

Contributed to OEIS on April 15, 2007 -- the 300th anniversary of the birth of Leonhard Euler.

References

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

Links

Table of n, a(n) for n=1..23.

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... = 1/1 - 1/(1*8) + 1/(1*8*13) - 1/(1*8*13*16) + 1/(1*8*13*16*64) - ...

Mathematica

nmax = 100; prec = 3000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Floor[ 1/(1 - #[[1]] #[[2]]) ], 1 - #[[1]] #[[2]]}&, {Floor[1/c], c}, nmax - 1]

Crossrefs

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

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

Sequence in context: A272486 A057486 A188198 * A338598 A168339 A070112

Adjacent sequences: A129407 A129408 A129409 * A129411 A129412 A129413

Keyword

nonn,easy

Author

Stuart Clary, Apr 15 2007

Status

approved