A129408 Continued fraction for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

0, 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 9, 4, 13, 4, 1, 2, 27, 1, 28, 1, 2, 2, 3, 2, 7, 1, 1, 19, 1, 8, 3, 3, 2, 1, 10, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 35, 1, 2, 91, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 16, 1, 2, 2, 1, 2, 6, 1, 1, 6, 14, 1, 5, 5, 14, 2, 8, 1, 1, 1, 1, 2, 4, 2, 10, 37, 1, 10, 2, 4, 5, 4, 5, 24, 1, 2, 7, 1

Offset

0,3

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=0..97.

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... = [0; 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 9, 4, 13, 4, ...].

Mathematica

nmax = 1000; ContinuedFraction[4 Pi^3/(81 Sqrt[3]), nmax + 1]

Crossrefs

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

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

Sequence in context: A117825 A010143 A101027 * A339748 A325470 A240831

Adjacent sequences: A129405 A129406 A129407 * A129409 A129410 A129411

Keyword

nonn,cofr,easy

Author

Stuart Clary, Apr 15 2007

Status

approved