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.

A339801 Decimal expansion of the real part of harmonic number H(1/2 + i*sqrt(3)/2), where i=sqrt(-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 17 2020

Keywords

Comments

For imaginary part see A339802.
For real b, Im(Psi(1/2 + b*i)) = Pi*tanh(Pi*b)/2, but no such closed formula is known for the real part (see Wikipedia link). - Vaclav Kotesovec, Dec 19 2020

Examples

			0.862289106171836386535085450...

Links

Crossrefs

Cf. A256919, A338815, A339135, A339529, A339530, A339604, A339605, A339606, A339802.

Programs

  • Mathematica
    RealDigits[N[Re[HarmonicNumber[1/2 + I Sqrt[3]/2]], 105]][[1]]

Formula

Equals 1/2 + gamma + Re(Psi(1/2 + i*sqrt(3)/2)), where gamma is the Euler-Mascheroni constant (see A001620) and Psi is the digamma function.
Equals -1/2 + 3*A339604 + 3*A339606.
Equals Re((1 + i*sqrt(3))*Sum_{k>=0} 1/((1 + k)*(3 + i*sqrt(3) + 2*k))).

A339802 Decimal expansion of the imaginary part of harmonic number H(1/2 + i*sqrt(3)/2) where i=sqrt(-1).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Dec 17 2020

Keywords

Comments

For real part of H(1/2 + i*sqrt(3)/2) see A339801.

Examples

			0.691215820928755403365848...

Crossrefs

Cf. A256919, A338815, A339135, A339529, A339530, A339605, A339605, A339606, A339801.

Programs

  • Mathematica
    RealDigits[N[Im[HarmonicNumber[1/2 + I Sqrt[3]/2]], 105]][[1]]

Formula

Equals (Pi/2)*tanh(Pi*sqrt(3)/2) - sqrt(3)/2.
Equals Im(Psi(3/2 + i*sqrt(3)/2)).
Equals -sqrt(3)/2 + Im(Psi(1/2 + i*sqrt(3)/2)).
Equals Im((1 + i*sqrt(3))*Sum_{k>=0} 1/((1 + k)*(3 + i*sqrt(3) + 2*k))).

A351432 Decimal expansion of sqrt(3)/2 + Pi*tanh(Pi*sqrt(3)/2)/2.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Feb 11 2022

Keywords

Comments

Imaginary part of psi(-1/2 + i*sqrt(3)/2) where psi is the digamma function.

Examples

			2.423266628497632696893294495...

Crossrefs

Cf. A024006, A100554, A113319, A256919, A259171, A338815, A339135, A339529, A339530, A339604, A339605, A339606, A347292.

Programs

  • Mathematica
    RealDigits[N[Sqrt[3]/2 + 1/2 Pi Tanh[Sqrt[3] Pi/2], 105]][[1]]
  • PARI
    imag(psi(-1/2+I*sqrt(3)/2)) \\ Michel Marcus, Feb 11 2022

Formula

Equals sqrt(3)*(1 - gamma/3 - Re(psi(-1/2 + i*sqrt(3)/2))/3 + A339606).

Extensions

Last two digits corrected by Georg Fischer, May 15 2024
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.