The On-Line Encyclopedia of Integer Sequences (OEIS)

The OEIS is supported by the many generous donors to the OEIS Foundation.

Revision History for A129409

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing all changes.

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

#9 by Michel Marcus at Fri Feb 13 02:24:34 EST 2015

STATUS

reviewed

approved

#8 by Joerg Arndt at Fri Feb 13 01:50:03 EST 2015

STATUS

proposed

reviewed

#7 by Michel Marcus at Fri Feb 13 01:30:05 EST 2015

STATUS

editing

proposed

#6 by Michel Marcus at Fri Feb 13 01:29:58 EST 2015

FORMULA

Series: L(3, chi3) = sum_{k >=1..infinity} 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 piPi^3/(81 sqrt(3)).

STATUS

proposed

editing

#5 by Jon E. Schoenfield at Fri Feb 13 00:38:29 EST 2015

STATUS

editing

proposed

#4 by Jon E. Schoenfield at Fri Feb 13 00:38:26 EST 2015

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

FORMULA

Closed form: L(3, chi3) = 4 pi^3/(81 sqrt(3)).

CROSSREFS

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

AUTHOR

Stuart Clary, Apr 15, 2007

STATUS

approved

editing

#3 by Russ Cox at Fri Mar 30 19:00:05 EDT 2012

AUTHOR

_Stuart Clary (clary(AT)uakron.edu), _, Apr 15, 2007

Discussion

Fri Mar 30

19:00

OEIS Server: https://oeis.org/edit/global/303

#2 by N. J. A. Sloane at Sun Jun 29 03:00:00 EDT 2008

COMMENTS

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

KEYWORD

nonn,easy,new

#1 by N. J. A. Sloane at Fri May 11 03:00:00 EDT 2007

NAME

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

DATA

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

1,1

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

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..infinity} 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/(2*2) + 1/(2*2*2) + 1/(2*2*2*14) + 1/(2*2*2*14*94) + ...

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]

CROSSREFS

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

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

KEYWORD

nonn,easy,new

AUTHOR

Stuart Clary (clary(AT)uakron.edu), Apr 15, 2007

STATUS

approved