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.

Previous Showing 11-13 of 13 results.

A261622 Decimal expansion of the Dirichlet beta function at 1/3.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Aug 27 2015

Keywords

Examples

			0.6178550888488520660725389947279931657106235478993865002255152822956...

Links

Crossrefs

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A261623 (beta(1/4)), A261624 (beta(1/5)).

Programs

  • Maple
    evalf(Sum((-1)^n/(2*n+1)^(1/3), n=0..infinity), 120); # Vaclav Kotesovec, Aug 27 2015
  • Mathematica
    RealDigits[DirichletBeta[1/3],10,105]//First
  • PARI
    beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
beta(1/3) \\ Charles R Greathouse IV, Oct 18 2024

Formula

beta(1/3) = (zeta(1/3, 1/4) - zeta(1/3, 3/4))/2^(2/3).

A261623 Decimal expansion of the Dirichlet beta function at 1/4.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Aug 27 2015

Keywords

Examples

			0.59072305644249473186595915351156205979836741723911440082771876593...

Links

Crossrefs

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A261622 (beta(1/3)), A261624 (beta(1/5)).

Programs

  • Maple
    evalf(Sum((-1)^n/(2*n+1)^(1/4), n=0..infinity), 120); # Vaclav Kotesovec, Aug 27 2015
  • Mathematica
    RealDigits[DirichletBeta[1/4],10,103]//First
  • PARI
    beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
beta(1/4) \\ Charles R Greathouse IV, Oct 18 2024

Formula

beta(1/4) = (zeta(1/4, 1/4) - zeta(1/4, 3/4))/sqrt(2).

A348763 Decimal expansion of Sum_{n>=1} ((-1)^(n+1)*n)/(n+1)^2.

Original entry on oeis.org

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

Views

Author

Dumitru Damian, Oct 31 2021

Keywords

Examples

			0.12931985286416790881897546186483602653397481624314396474709910519161011...

Links

Crossrefs

Cf. A006752, A072691, A111003, A153071, A175571, A181983, A300707.

Programs

  • Mathematica
    RealDigits[Pi^2/12 - Log[2], 10, 100][[1]] (* Amiram Eldar, Nov 30 2021 *)
  • PARI
    -sumalt(n=1, (-1)^n*n/(n+1)^2) \\ Charles R Greathouse IV, Nov 01 2021
  • PARI
    Pi^2/12-log(2) \\ Charles R Greathouse IV, Nov 01 2021
  • Python
    from scipy.special import zeta
from math import log
int(''.join(n for n in list(str(zeta(2)/2-log(2)))[2:-2]))
  • Python
    int(str(sum((-1)**(n+1)*n/(n+1)**2 for n in range(1,5000000)))[2:-2])
  • SageMath
    (pi^2/12-log(2)).n(digits=100)

Formula

Equals Pi^2/12-log(2).
Equals Sum_{k>=2} (zeta(k)-zeta(k+1))/2^k. - Amiram Eldar, Mar 20 2022
Equals Integral_{x >= 0} x/(1 + exp(x))^2 dx = (1/2) * Integral_{x >= 0} x*(x - 2)*exp(x)/(1 + exp(x))^2 dx . - Peter Bala, Apr 26 2025
Previous Showing 11-13 of 13 results.

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