login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A086724 Decimal expansion of g(1)-g(2)+g(4)-g(5), where g(k) = sum(1/(6*m+k)^2, m >= 0). 12
7, 8, 1, 3, 0, 2, 4, 1, 2, 8, 9, 6, 4, 8, 6, 2, 9, 6, 8, 6, 7, 1, 8, 7, 4, 2, 9, 6, 2, 4, 0, 9, 2, 3, 5, 6, 3, 6, 5, 1, 3, 4, 3, 3, 6, 5, 4, 5, 2, 8, 5, 4, 2, 0, 2, 2, 2, 1, 0, 0, 0, 6, 2, 9, 6, 6, 8, 8, 6, 9, 8, 4, 6, 5, 1, 6, 1, 8, 2, 1, 8, 0, 9, 2, 8, 6, 9, 5, 7, 0, 8, 3, 2, 2, 0, 9, 8, 6, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This number is L(2, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3, A102283. - Stuart Clary, Dec 17 2008

Equals 1/1^2 -1/2^2 +1/4^2 -1/5^2 +1/7^2 -1/8^2 +1/10^2 -1/11^2 +-... . This can be split as (1/1^2 -1/5^2 +1/7^2 -1/11^2 +-...) - (1/2^2 -1/4^2 +1/8^2 -1/10^2..) = (g(1)-g(5)) - (g(2)-g(4)). The first of these two series is A214552 and the second series is 1/(2^2)*(1-1/2^2 +1/4^2-1/5^2+-...), namely a quarter of the original series. Therefore 5/4 of this value here equals A214552. - R. J. Mathar, Jul 20 2012

REFERENCES

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

LINKS

Table of n, a(n) for n=0..98.

D. H. Bailey, J. M. Borwein, R. E. Crandall, Integrals of the Ising class, J. Phys. A 39 (2006) 12271, variable C_3.

K. H. Pilehrood, T. H. Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, El. J. Combin. 18 (2) (2012), #P35, value of K after Theorem 4.

FORMULA

sum(jacobi(-3, n+3)/n^2, n >= 1), also equals (8/15)*4F3(1/2,1,1,2; 5/4,3/2,7/4; 3/4), where 4F3 is the generalized hypergeometric function, or also 4*Pi*log(3)/(3*sqrt(3)) - 4*integral_(0..1) log(x+1)/(x^2-x+1) dx. - Jean-Fran├žois Alcover, Jul 17 2014, updated Jan 23 2015

EXAMPLE

0.781302412896486296867...

MATHEMATICA

nmax = 1000; First[ RealDigits[(Zeta[2, 1/3] - Zeta[2, 2/3])/9, 10, nmax] ] (* Stuart Clary, Dec 17 2008 *)

PROG

(PARI) zetahurwitz(2, 1/3)/9 - zetahurwitz(2, 2/3)/9 \\ Charles R Greathouse IV, Jan 30 2018

CROSSREFS

Cf. A086722-A086731.

Cf. A153066, A153067, A153068. - Stuart Clary, Dec 17 2008

Sequence in context: A256670 A021132 A019936 * A268979 A329219 A093720

Adjacent sequences:  A086721 A086722 A086723 * A086725 A086726 A086727

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Jul 31 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 29 10:06 EDT 2020. Contains 337428 sequences. (Running on oeis4.)