A354052 -id:A354052 - 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))).

A354053 Decimal expansion of Sum_{k>=0} 1 / (k^8 + 1).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 16 2022

			1.504062133314799511292905417451127075245414363820351975458635357818812...

Cf. A060890, A113319, A354051, A354052.

evalf(1/2 + ((sqrt(2 + sqrt(2))*sinh(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sin(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi)) + (sqrt(2 + sqrt(2))*sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi))) * Pi/8, 100);

RealDigits[Chop[N[Sum[1/(k^8 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^8 + 1))
```

Equals 1/2 + ((sqrt(2 + sqrt(2))*sinh(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sin(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi)) + (sqrt(2 + sqrt(2))*sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi))) * Pi/8.

Equal 3/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(8*k)-1). - Amiram Eldar, May 20 2022

A354054 a(n) = Product_{k=0..n} (k^6 + 1).

Original entry on oeis.org

1, 2, 130, 94900, 388805300, 6075471617800, 283463279271694600, 33349454806314869690000, 8742392830201411514885050000, 4646074730467898538293540742100000, 4646079376542629006192079035640742100000, 8230817672466612927467651920537784356160200000

Offset: 0

Vaclav Kotesovec, May 16 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..103

Cf. A002604, A101686, A255433, A255434, A258871, A354052.

A354054 := proc(n)
    mul( k^6+1,k=0..n) ;
end proc:
seq(A354054(n),n=0..40) ; # R. J. Mathar, Jul 17 2023

Table[Product[k^6 + 1, {k, 0, n}], {n, 0, 15}]

PARI
```
a(n) = prod(k=1, n, k^6+1);
```

a(n) ~ (2*sinh(2*Pi) - 4*sinh(Pi)*cos(sqrt(3)*Pi)) * n^(6*n + 3) / exp(6*n).

a(n) ~ A258871 * (n!)^6.

cp's OEIS Frontend

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

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A354053 Decimal expansion of Sum_{k>=0} 1 / (k^8 + 1).

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A354054 a(n) = Product_{k=0..n} (k^6 + 1).

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