A099218 -id:A099218 - cp's OEIS frontend

A255685 Decimal expansion of the alternating double sum U(3,1) = Sum_{i>=2} (Sum_{j=1..i-1} (-1)^(i+j)/(i^3*j)) (negated).

1, 1, 7, 8, 7, 5, 9, 9, 9, 6, 5, 0, 5, 0, 9, 3, 2, 6, 8, 4, 1, 0, 1, 3, 9, 5, 0, 8, 3, 4, 1, 3, 7, 6, 1, 8, 7, 1, 5, 2, 1, 7, 5, 1, 3, 1, 7, 5, 9, 7, 5, 0, 6, 3, 3, 2, 2, 2, 4, 5, 2, 4, 1, 8, 5, 4, 2, 7, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 4, 1, 3, 2, 4, 3, 7, 0, 1, 7, 4, 6, 4, 8, 2, 7, 1, 2, 5, 9, 5, 1, 3, 2, 4

Offset: 0

Jean-François Alcover, Mar 02 2015

			-0.117875999650509326841013950834137618715217513175975...

David Broadhurst, Feynman’s sunshine numbers, arXiv:1004.4238 [physics.pop-ph], 2010, p. 16.
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 6.

Cf. A099218.

U[3,1] = Pi^4/180 + (Pi^2/12)*Log[2]^2  - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2]; RealDigits[U[3,1], 10, 103] // First

Pi^4/180 + (Pi^2/12)*log(2)^2  - (1/12)*log(2)^4 - 2*polylog(4, 1/2) \\ Gheorghe Coserea, Sep 30 2018

Pi^4/180 + (Pi^2/12)*log(2)^2 - (1/12)*log(2)^4 - 2*Li_4(1/2).

A099217 Decimal expansion of Li_3(1/2).

Original entry on oeis.org

5, 3, 7, 2, 1, 3, 1, 9, 3, 6, 0, 8, 0, 4, 0, 2, 0, 0, 9, 4, 0, 6, 2, 3, 2, 2, 5, 5, 9, 4, 9, 6, 5, 8, 2, 6, 6, 7, 0, 4, 0, 2, 4, 9, 9, 3, 4, 0, 3, 7, 8, 1, 7, 0, 6, 8, 9, 7, 6, 1, 9, 3, 0, 7, 1, 8, 3, 2, 4, 0, 8, 0, 9, 2, 0, 1, 3, 8, 3, 9, 7, 3, 3, 0, 4, 1, 2, 3, 5, 9, 9, 7, 5, 4, 3, 9, 6, 7, 0, 0, 4, 8, 1, 4

Offset: 0

Benoit Cloitre, Oct 06 2004

			0.537213193608040200940623225594965826670402499340...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 44.
L. Lewin, Polylogarithms and associated Functions, North Holland (1981) A.2.6(3)

R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I, Il Nuovo Cim. 11A (4) (1972) 824-864, variable a_3 and table II (15).
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 473.

Cf. A076788, A099218, A099219, A099220, A099221, A099222, A099223, A099224.

RealDigits[PolyLog[3, 1/2], 10, 104] // First (* Jean-François Alcover, Feb 13 2013 *)

polylog(3,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_3(1/2) = Sum_{k>0} 1/(2^k*k^3) = 0.537213193608...

Li_3(1/2) = 7*zeta(3)/8-Pi^2*log(2)/12+log(2)^3/6. - Benoit Cloitre, May 22 2006

Leading zero removed by R. J. Mathar, Feb 05 2009

A099219 Decimal expansion of Li_5(1/2).

Original entry on oeis.org

5, 0, 8, 4, 0, 0, 5, 7, 9, 2, 4, 2, 2, 6, 8, 7, 0, 7, 4, 5, 9, 1, 0, 8, 8, 4, 9, 2, 5, 8, 5, 8, 9, 9, 4, 1, 3, 1, 9, 5, 4, 1, 1, 2, 5, 6, 6, 4, 8, 2, 1, 6, 4, 8, 7, 2, 4, 4, 9, 7, 7, 9, 6, 3, 5, 2, 6, 2, 5, 3, 9, 4, 2, 2, 8, 7, 8, 0, 2, 4, 2, 6, 1, 9, 3, 8, 4, 2, 1, 0, 0, 4, 9, 3, 4, 4, 9, 5, 5, 0, 6, 2, 2, 5

Offset: 0

Benoit Cloitre, Oct 06 2004

			0.50840057924...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch (1996), Section 1.3.2.

Cf. A076788, A099217, A099218, A099220, A099221, A099222, A099223, A099224.

polylog(5,1/2) ; evalf(%) ; # R. J. Mathar, Feb 15 2013

RealDigits[N[PolyLog[5, 1/2], 104]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(5,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_5(1/2) = Sum_{k>0} 1/2^k/k^5.

A099220 Decimal expansion of Li_6(1/2).

Original entry on oeis.org

5, 0, 4, 0, 9, 5, 3, 9, 7, 8, 0, 3, 9, 8, 8, 5, 5, 0, 6, 9, 0, 0, 4, 6, 5, 0, 9, 7, 8, 8, 8, 7, 9, 0, 9, 5, 2, 0, 6, 5, 2, 2, 2, 8, 9, 3, 2, 6, 6, 7, 4, 4, 4, 9, 2, 3, 1, 3, 4, 5, 2, 6, 1, 0, 8, 3, 6, 3, 5, 9, 2, 0, 8, 5, 5, 6, 6, 7, 5, 5, 8, 4, 4, 4, 3, 3, 9, 8, 6, 8, 7, 6, 2, 1, 1, 7, 1, 2, 5, 0, 7, 6, 3, 2

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099221, A099222, A099223, A099224.

RealDigits[ N[ PolyLog[6, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(6,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_6(1/2)=sum(k>0, 1/2^k/k^6)=0.5040953978...

A099221 Decimal expansion of Li_7(1/2).

Original entry on oeis.org

5, 0, 2, 0, 1, 4, 5, 6, 3, 3, 2, 4, 7, 0, 8, 4, 9, 4, 5, 6, 7, 4, 8, 9, 2, 9, 5, 6, 4, 0, 7, 0, 7, 0, 3, 6, 2, 8, 0, 1, 8, 8, 1, 5, 2, 4, 8, 9, 7, 8, 8, 7, 3, 4, 2, 7, 2, 1, 7, 5, 6, 5, 9, 5, 6, 3, 5, 4, 5, 0, 6, 6, 3, 5, 9, 1, 9, 9, 3, 1, 9, 6, 5, 0, 7, 8, 3, 2, 0, 5, 0, 4, 9, 0, 9, 4, 8, 4, 7, 7, 9, 6, 8, 6

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099220, A099222, A099223, A099224.

RealDigits[ N[ PolyLog[7, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(7,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_7(1/2)=sum(k>0, 1/2^k/k^7)=0.50201456332470849456....

Formula value corrected and leading zero removed by R. J. Mathar, Feb 05 2009

A099222 Decimal expansion of Li_8(1/2).

Original entry on oeis.org

5, 0, 0, 9, 9, 6, 6, 5, 9, 0, 9, 7, 0, 5, 1, 9, 1, 0, 5, 5, 7, 3, 5, 5, 9, 0, 5, 5, 3, 0, 2, 7, 2, 4, 5, 8, 7, 2, 5, 9, 5, 5, 8, 3, 5, 8, 8, 8, 6, 0, 5, 2, 1, 7, 5, 3, 0, 8, 6, 6, 4, 8, 3, 4, 1, 7, 6, 9, 1, 7, 8, 7, 7, 3, 2, 4, 3, 0, 4, 9, 7, 4, 0, 0, 2, 8, 2, 4, 1, 0, 7, 4, 5, 3, 9, 9, 8, 2, 2, 9, 4, 0, 8, 1

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099220, A099221, A099223, A099224.

RealDigits[ N[ PolyLog[8, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(8,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_8(1/2)=sum(k>0, 1/2^k/k^8)=0.500996659...

A099223 Decimal expansion of Li_9(1/2).

Original entry on oeis.org

5, 0, 0, 4, 9, 4, 8, 8, 8, 1, 0, 5, 9, 5, 3, 6, 1, 0, 0, 4, 0, 4, 8, 6, 0, 1, 6, 4, 1, 2, 9, 5, 2, 3, 6, 3, 0, 7, 0, 2, 3, 6, 0, 5, 1, 2, 8, 1, 6, 9, 7, 1, 7, 6, 5, 2, 5, 4, 0, 4, 6, 6, 0, 2, 1, 9, 8, 0, 2, 7, 9, 4, 5, 6, 1, 9, 5, 9, 4, 5, 0, 0, 2, 3, 9, 8, 9, 1, 0, 2, 4, 0, 6, 8, 4, 4, 3, 2, 1, 2, 0, 5, 1, 5

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099220, A099221, A099222, A099224.

RealDigits[ N[ PolyLog[9, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(9,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_9(1/2)=sum(k>0, 1/2^k/k^9)=0.5004948881...

A099224 Decimal expansion of Li_{10}(1/2).

Original entry on oeis.org

5, 0, 0, 2, 4, 6, 3, 2, 0, 6, 0, 6, 0, 0, 6, 7, 7, 5, 0, 0, 9, 6, 7, 5, 2, 4, 0, 4, 9, 6, 0, 2, 7, 5, 5, 5, 3, 3, 4, 4, 1, 1, 3, 1, 0, 4, 3, 7, 1, 7, 3, 9, 0, 7, 8, 6, 4, 7, 2, 4, 7, 9, 7, 7, 6, 4, 1, 5, 5, 6, 1, 8, 4, 4, 6, 5, 0, 0, 1, 6, 5, 5, 5, 9, 2, 4, 1, 5, 5, 2, 0, 6, 7, 2, 4, 0, 7, 3, 9, 1, 8, 6, 8, 5

Offset: 0

Benoit Cloitre, Oct 06 2004

Cf. A076788, A099217, A099218, A099219, A099220, A099221, A099222, A099223.

RealDigits[ N[ PolyLog[10, 1/2], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

polylog(10,1/2) \\ Charles R Greathouse IV, Jul 14 2014

Li_{10}(1/2)=sum(k>0, 1/2^k/k^10)=0.500246320606006775...

A074903 Decimal expansion of the mean number of iterations in comparing two numbers via their continued fractions.

Original entry on oeis.org

1, 3, 5, 1, 1, 3, 1, 5, 7, 4, 4, 9, 1, 6, 5, 9, 0, 0, 1, 7, 9, 3, 8, 6, 8, 0, 0, 5, 2, 5, 6, 5, 2, 1, 0, 6, 8, 3, 6, 0, 6, 5, 1, 5, 0, 8, 7, 4, 2, 7, 0, 1, 6, 8, 7, 3, 4, 5, 1, 4, 7, 2, 1, 1, 0, 1, 3, 7, 4, 2, 2, 7, 7, 1, 1, 9, 5, 5, 0, 1, 7, 1, 2, 8, 6, 9, 1, 3, 0, 7, 5, 1, 5, 9, 7, 8, 0, 2, 3, 9

Offset: 1

N. J. A. Sloane, Sep 15 2002

Another description: Decimal expansion of the mean number of comparisons (moment sum of index 2) in the basic continued fraction sign algorithm ("BCF-sign").

Still another description: Decimal expansion of expected number of iterations of Gaussian reduction of a 2-dimensional lattice.

			1.351131574491659001793868005256521068360651508742701687345147211...
(Only the first 31 digits are the same as those given by Flajolet & Vallée. - _Jean-François Alcover_, Apr 23 2015)

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 161.
Philippe Flajolet and Brigitte Vallée, Continued fraction algorithms and constants, in "Constructive, Experimental, and Nonlinear Analysis", Michel Théra Editor, CMS Conference Proceedings, Canadian Mathematical Society Volume 27 (2000), p. 67.

H. Daude, P. Flajolet, and B. Vallee, An average-case analysis of the Gaussian algorithm for lattice reduction, INRIA, 1996. [alternative link]
Philippe Flajolet, Continued Fractions, Comparison Algorithms and Fine Structure Constants.
Eric Weisstein's World of Mathematics, Polylogarithm.
Eric Weisstein's World of Mathematics, Vallée Constant.

Cf. A099218.

17 - 60/Pi^4 (24*PolyLog[4, 1/2] - Pi^2*Log[2]^2 + 21*Zeta[3]*Log[2] + Log[2]^4) // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Mar 19 2013, after Steven Finch *)

17 - 60*(24*polylog(4, 1/2) - Pi^2*log(2)^2 + 21*zeta(3)*log(2) + log(2)^4)/Pi^4 \\ Charles R Greathouse IV, Aug 27 2014

Equals (-60/Pi^4)*(24*Li_4(1/2) - Pi^2*log(2)^2 + 21*zeta(3)*log(2) + log(2)^4) + 17, with Li_4 the tetralogarithm function. - Jean-François Alcover, Apr 23 2015

Corrected and extended by Jean-François Alcover, Mar 19 2013

Entry revised by N. J. A. Sloane, Apr 24 2015 to include information from two other entries (submitted respectively by Eric W. Weisstein, Aug 05 2008 and Jean-François Alcover, Apr 23 2015) that formerly described this same constant.

A214508 Decimal expansion of the series limit sum_{k>=1} (-1)^(k+1) sum_{t=1..k} 1/(t^2*(k+1)^2).

Original entry on oeis.org

1, 6, 2, 6, 5, 4, 6, 6, 7, 3, 9, 7, 4, 2, 0, 0, 8, 0, 7, 7, 5, 5, 6, 4, 5, 6, 5, 1, 7, 3, 5, 9, 1, 1, 0, 1, 1, 8, 7, 0, 6, 4, 2, 0, 8, 3, 3, 7, 6, 5, 9, 9, 2, 3, 7, 2, 6, 7, 6, 3, 0, 6, 9, 8, 3, 1, 8, 4, 3, 5, 7, 7, 2, 9, 8, 2, 1, 0, 7, 4, 9, 2, 1, 6, 7, 2, 0, 0, 7, 4, 6, 3, 7, 5, 7, 4, 9, 8, 1, 0, 6, 7, 9, 6, 9

Offset: 0

R. J. Mathar, Jul 19 2012

Equals the alternating sum over (-1)^(k+1)*H_k^(2)/(k+1)^2, where H_k^(2) is the harmonic sum over inverse squares, H_k^(2) = sum_{t=1..k} 1/t^2 = 1, 5/4, 49/36, 205/144, 5269/3600,..., see A007406. The sum over H_k^(2)/(k+1)^2, over the absolute values, is Pi^4/120 = 0.811742425283353...

			0.162654667397420080...

D. H. Bailey, J. M. Borwein, R. Girgensohn, Experimental evaluation of Euler sums, Exp. Math. 3 (1994) 17, variable alpha(2,2)
G. Rutledge, R. D. Douglass, Table of definite integrals, Am. Math. Monthly 45 (1938) 525, variable A_4.

a099218 := polylog(4,1/2) ;
-4*a099218+13*Pi^4/288-7/2*Zeta(3)*log(2)+Pi^2/6*(log(2))^2-(log(2))^4/6 ;
evalf(%) ;

NSum[(-1)^(k + 1)*HarmonicNumber[k, 2]/(k + 1)^2, {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 110] // RealDigits[#, 10, 105] & // First (* or, from formula: *) 13*Pi^4/288 + 1/6*Pi^2*Log[2]^2 - 1/6*Log[2]*(Log[2]^3 + 21*Zeta[3]) - 4*PolyLog[4, 1/2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 06 2013 *)

13*Pi^4/288 + 1/6*Pi^2*log(2)^2 - 1/6*log(2)*(log(2)^3 + 21*zeta(3)) - 4*polylog(4, 1/2) \\ Charles R Greathouse IV, Jul 18 2014

Equals -4*A099218 +13*Pi^4/288 -7*A002117*log(2)/2+log^2(2)*(Pi^2-log^2(2))/6.

More terms from Jean-François Alcover, Feb 12 2013

cp's OEIS Frontend

A255685 Decimal expansion of the alternating double sum U(3,1) = Sum_{i>=2} (Sum_{j=1..i-1} (-1)^(i+j)/(i^3*j)) (negated).

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A099217 Decimal expansion of Li_3(1/2).

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A099219 Decimal expansion of Li_5(1/2).

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A099220 Decimal expansion of Li_6(1/2).

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A099221 Decimal expansion of Li_7(1/2).

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A099222 Decimal expansion of Li_8(1/2).

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A099223 Decimal expansion of Li_9(1/2).

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A099224 Decimal expansion of Li_{10}(1/2).

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