A129409 Engel expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

2, 2, 2, 14, 94, 372, 1391, 7690, 17729, 49204, 87816, 128433, 151275, 290477, 297212, 299837, 352249, 897751, 1081032, 1646358, 2402614, 36591866, 49132456, 93538655, 141789387, 180474393, 687775235, 851204316, 1868593596, 7042652755

Offset: 1

Author

Stuart Clary, Apr 15 2007

Keywords

• nonn
• easy

Comments

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

Examples

			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) + ...
		

References

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

Crossrefs

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

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

Programs

Mathematica

nmax = 100; prec = 2000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Ceiling[1/(#[[1]] #[[2]] - 1)], #[[1]] #[[2]] - 1}&, {Ceiling[1/c], c}, nmax - 1]

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)).