A354053 -id:A354053 - cp's OEIS frontend

A354051 Decimal expansion of Sum_{k>=0} 1 / (k^4 + 1).

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

Offset: 1

Vaclav Kotesovec, May 16 2022, following a suggestion from Bernard Schott

Apart from leading digits the same as A256920. - R. J. Mathar, May 20 2022

			1.578477579667136838318022193245719235046672217327291327587486645793808...

Cf. A256920, A113319, A354052, A354053.

Cf. A002523, A255434, A354004.

evalf(1/2 + (Pi*(sinh(sqrt(2)*Pi) + sin(sqrt(2)*Pi))) / (2*sqrt(2)*(cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi))), 105);

RealDigits[Chop[N[Sum[1/(k^4 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^4 + 1))
```

Equals 1/2 + (Pi*(sinh(sqrt(2)*Pi) + sin(sqrt(2)*Pi))) / (2*sqrt(2)*(cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi))).

A354052 Decimal expansion of Sum_{k>=0} 1 / (k^6 + 1).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 16 2022

			1.517100734033216426152907644902413858062211322529844672847634899037901...

Cf. A002604, A113319, A354051, A354053.

evalf(1/2 + (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi)))*Pi/6, 100);

RealDigits[Chop[N[Sum[1/(k^6 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^6 + 1))
```

Equals 1/2 + (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi)))*Pi/6.

Equal 3/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(6*k)-1). - Amiram Eldar, May 20 2022

cp's OEIS Frontend

A354051 Decimal expansion of Sum_{k>=0} 1 / (k^4 + 1).

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