A129411 Greedy Egyptian expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

2, 3, 20, 1449, 2879423, 31625640285294, 1162849840832612010600369938, 4013794377413687199924671384130798842309412001723286013, 32025095658857878502181254937184611855940944199483548417530154807379258429933254996925647878294253643673560013

Offset

1,1

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

Links

Table of n, a(n) for n=1..9.

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 + 1/3 + 1/20 + 1/1449 + 1/2879423 + ...

Mathematica

nmax = 12; prec = 2000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Ceiling[1/(#[[2]] - 1/#[[1]])], #[[2]] - 1/#[[1]]}&, {Ceiling[1/c], c}, nmax - 1]

Crossrefs

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

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

Sequence in context: A292866 A132421 A132500 * A124447 A250181 A024765

Adjacent sequences: A129408 A129409 A129410 * A129412 A129413 A129414

Keyword

nonn,easy

Author

Stuart Clary, Apr 15 2007

Status

approved