cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A355234 Decimal expansion of Li_2(-1/2), the dilogarithm of (-1/2) (negated).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jun 25 2022

Keywords

Examples

			-0.44841420692364620244306440591577432083426994134919...

Links

Crossrefs

Cf. A001008, A002805, A007758.
Other values of Li_2: A072691, A076788, A152115, A242599, A242600.

Programs

  • Mathematica
    RealDigits[PolyLog[2, -1/2], 10, 100][[1]]
  • PARI
    -dilog(-1/2) \\ Michel Marcus, Jun 25 2022

Formula

From Shamos (2011):
Equals -Li_2(1/3) - log(3/2)^2/2.
Equals Li_2(2/3) + log(3)^2/2 - log(2)^2/2 - Pi^2/6.
Equals Li_2(1/4)/2 + log(2)^2/2 - Pi^2/12.
Equals -Sum_{k>=1} (-1)^(k+1)/(2^k*k^2) = -Sum_{k>=1} (-1)^(k+1)/A007758(k).
Equals -Sum_{k>=1} H(k)/(k*3^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals -Integral_{x=0..1} log(x)^2/(x+2)^2 dx.
Equals -Integral_{x>=1} log(x)^2/(2*x+1)^2 dx.
Equals Integral_{x=0..1} log(x)/(x+2) dx.
Equals -Integral_{x>=0} log(1 + exp(-x)/2) dx.

A242599 Decimal expansion of dilog(phi-1) = polylog(2, 2-phi) with phi = (1 + sqrt(5))/2.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Jun 16 2014

Keywords

Comments

dilog(phi-1) = polylog(2, 2-phi) = sum((2-phi)^k/k^2 , k =1 ..infinity) = sum((1 - 2*sin(Pi/10))^(2*k)/k^2, k=1..infinity) = Pi^2/15 - (log(phi-1))^2 = Pi^2/15 - (2/5)*log(phi-1)*(log(2-phi) + log(phi-1)/2).
See the Jolley reference pp. 66-69, (360)(e), and the Abramowitz-Stegun link, p. 1004, eqs. 27.7.3 - 27.7.6 with x = phi-1, solving for dilog(x) = f(x).

Examples

			0.42640880616209618209...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961.

Links

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A152115, A076788 (dilog(1/2)), A242600.

Programs

  • Maple
    phi := (1+sqrt(5))/2 ; dilog(phi-1) ; evalf(%) ; # R. J. Mathar, Jun 10 2024
  • Mathematica
    RealDigits[PolyLog[2, 2 - GoldenRatio], 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)
  • PARI
    polylog(2, 2 - (1+sqrt(5))/2) \\ Gheorghe Coserea, Sep 30 2018
  • PARI
    sumpos(k=1, (1 - 2*sin(Pi/10))^k/k^2) \\ Gheorghe Coserea, Sep 30 2018

Formula

dilog(phi-1) = polylog(2, 2-phi) = Sum_{k>=1} (2-phi)^k/k^2 = Sum_{k>=1} (1 - 2*sin(Pi/10))^k/k^2.
Showing 1-2 of 2 results.

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