A100554 -id:A100554 - cp's OEIS frontend

A113319 Decimal expansion of Sum_{k>=0} 1/(k^2+1).

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

Offset: 1

Benoit Cloitre, Jan 07 2006

Known to be transcendental. After n=2 it is the same as A100554(n).

Imaginary part of psi(I) (for the real part, see A248177). - Stanislav Sykora, Oct 03 2014

			2.076674047468581174134050794750000490445656266403816665575062484390...

Michel Waldschmidt, Elliptic functions and transcendance, Surveys in number theory, 143-188, Dev. Math., 17, Springer, New York, 2008.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Digamma function.
Index entries for transcendental numbers.

Cf. A013661 (Sum_{i>=1} 1/i^2), A232883 (Sum_{i>=0} 1/(2*i^2+1)). - Bruno Berselli, Dec 02 2013
Cf. A248177.
Essentially the same as A100554 and A259171.

RealDigits[N[Im[PolyGamma[0,I]],105]][[1]]  (* Vaclav Kotesovec, Oct 03 2014 *)

PARI
```
1/2+Pi/tanh(Pi)/2
```

imag(psi(I)) \\ Stanislav Sykora, Oct 03 2014

sumnumrat(1/(x^2+1),0) \\ Charles R Greathouse IV, Jan 20 2022

Equals 1/2 + Pi /(2*tanh(Pi)).
Equals 1+Integral_{x >= 0} sin(x)/(exp(x)-1) dx. - Robert FERREOL, Jan 12 2016.
Equals Sum_{k>=0} (-1)^(k+1)*(zeta(2*k) - 1). - Amiram Eldar, Apr 28 2025

Offset changed from 0 to 1 by Bruno Berselli, Dec 02 2013

A259171 Decimal expansion of Sum_{m>=1} 1/(m^2 + 1).

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Jun 19 2015

Essentially the same as A100554 and A113319. - R. J. Mathar, Jul 06 2015

			1.07667404746858117413405079475000049044565626640381666557

Muniru A Asiru, Table of n, a(n) for n = 1..3000
Junyong Zhao, Shaofang Hong, and Xiao Jiang, A certain reciprocal power sum is never an integer, arXiv:1812.08705 [math.NT], 2018. See the constant alpha_f.

Cf. A013661, A259173.

evalf[120]((Pi*coth(Pi)-1)/2); # Muniru A Asiru, Dec 21 2018

N[Sum[1/(k^2+1),{k,Infinity}],1000]//RealDigits//First

(Pi*cosh(Pi)/sinh(Pi)-1)/2 \\ Michel Marcus, Jun 28 2015

sumnumrat(1/(x^2+1), 1) \\ Charles R Greathouse IV, Jan 20 2022

Equals (Pi*coth(Pi)-1)/2. - Vaclav Kotesovec, Jun 27 2015
Equals Integral_{x>=0} sin(x)/(exp(x) - 1) dx. - Amiram Eldar, Aug 16 2020
Equals Integral_{x>=0} (sin(x)/sinh(x))^2 dx. - Amiram Eldar, Dec 11 2023

A256919 Decimal expansion of Sum_{k>=1} (zeta(4*k) - 1).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.0866629762657094129329746026249997547771718667980916672...
= -3 + Pi^4/90 + Pi^8/9450 + 691*Pi^12/638512875 + ...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights (2011) p. 265.

V. S. Adamchik, H. M. Srivastava, Some series of Zeta and related functions, Analysis (Munich) 18 (2) (1998) 131-144, eq. (2.25).
Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A013662, A013666, A013670, A175316, A256920, A339529, A339530, A339604 (3k-1).

Join[{0}, RealDigits[7/8 - (Pi/4)*Coth[Pi], 10, 104] // First]

Equals 7/8 - (Pi/4)*coth(Pi).
Equals Sum_{n>=2} 1/(n^4 - 1). - Vaclav Kotesovec, Dec 08 2020
Equals (1/2)* Sum_{n>=2} 1/(n^2-1) - (1/2)* Sum_{n>=2} 1/(n^2+1) = (3/4 - A100554)/2. - R. J. Mathar, Jan 22 2021

A351432 Decimal expansion of sqrt(3)/2 + Pitanh(Pisqrt(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

Artur Jasinski, Feb 11 2022

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

			2.423266628497632696893294495...

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

RealDigits[N[Sqrt[3]/2 + 1/2 Pi Tanh[Sqrt[3] Pi/2], 105]][[1]]

imag(psi(-1/2+I*sqrt(3)/2)) \\ Michel Marcus, Feb 11 2022

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

Last two digits corrected by Georg Fischer, May 15 2024

cp's OEIS Frontend

A113319 Decimal expansion of Sum_{k>=0} 1/(k^2+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A259171 Decimal expansion of Sum_{m>=1} 1/(m^2 + 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A256919 Decimal expansion of Sum_{k>=1} (zeta(4*k) - 1).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A351432 Decimal expansion of sqrt(3)/2 + Pitanh(Pisqrt(3)/2)/2.