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

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

1,2

Links

Table of n, a(n) for n=1..105.

Formula

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

Example

1.504062133314799511292905417451127075245414363820351975458635357818812...

Maple

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);

Mathematica

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

Prog

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

Crossrefs

Cf. A060890, A113319, A354051, A354052.

Sequence in context: A175296 A076266 A350281 * A200102 A016581 A175472

Adjacent sequences: A354050 A354051 A354052 * A354054 A354055 A354056

Keyword

nonn,cons

Author

Vaclav Kotesovec, May 16 2022

Status

approved