A129407 Balanced ternary expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

1, 0, -1, 0, 0, -1, -1, 1, 1, 0, 1, -1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1, 0, -1, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 1, 1, 0, 0, 1, 1, 1, 1, -1, 1, 1, 1, 0, 1, 0, -1, 0, 0, 0, 1, -1, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 1, 0, -1, -1, 0, 0, -1, 0, 1, 0, -1, 1, 1, -1, 1, -1, 0, 0, 0, -1, -1, 1, 1, 1, 1, 1, 0, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1

Offset

0,1

Comments

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

References

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292 (for this constant); Articles 330 and 331 (for balanced ternary)

Links

Table of n, a(n) for n=0..105.

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 + 0/3 - 1/3^2 + 0/3^3 + 0/3^4 - 1/3^5 - 1/3^6 + 1/3^7 + 1/3^8 + ...

Mathematica

nmax = 1000; prec = nmax/2 + 20 (* Normally this is sufficient precision. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Round[3(#[[2]] - #[[1]])], 3(#[[2]] - #[[1]])}&, {Round[c], c}, nmax]

Crossrefs

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

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

Sequence in context: A071982 A266588 A157749 * A267594 A273512 A266619

Adjacent sequences: A129404 A129405 A129406 * A129408 A129409 A129410

Keyword

sign,easy

Author

Stuart Clary, Apr 15 2007

Status

approved