A342361 -id:A342361 - cp's OEIS frontend

A342359 Decimal expansion of arctan(sqrt(Omega)), where Omega=LambertW(1) is the Omega constant.

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

Offset: 0

Gleb Koloskov, Mar 09 2021

The sine and the cosine of this angle appears in the values of two definite integrals that involve non-principal real branch of the Lambert W function, see A342360 and A342361.

			0.6454752445650039244357315545660663652246772055940215161816...

Cf. A342360, A342361, A030178.

Omega=LambertW[1]; xi=ArcTan[Sqrt[Omega]]; N[xi,120]

PARI
```
atan(sqrt(lambertw(1)))
```

Equals arctan(sqrt(LambertW(1))).

A342360 Decimal expansion of 1/(Omega+1)^2, where Omega=LambertW(1) is the Omega constant.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 09 2021

			0.40717638729656715790289020473539767731...

Cf. A342359, A342361, A030178, A030797.

Omega=LambertW[1]; xi=ArcTan[Sqrt[Omega]]; N[Cos[xi]^4,120]
Omega=LambertW[1]; N[1/(Omega+1)^2,120]
Omega=LambertW[1]; omega=1/Omega; NIntegrate[(-t/LambertW[-1,-t*Omega^omega])^Omega,{t,0,1}, WorkingPrecision->120]

PARI
```
cos(atan(sqrt(lambertw(1))))^4
```
PARI
```
my(Omega=lambertw(1)); 1/(Omega+1)^2
```

Equals cos(A342359)^4 = 1/(A030178+1)^2 = (1-sqrt(A342361))^2.
Equals Integral_{t=0..1} (-t/LambertW(-1,-t*Omega^omega))^Omega, where omega=1/Omega=1/LambertW(1).
Equals A115287^2. - Vaclav Kotesovec, Mar 12 2021

cp's OEIS Frontend

A342359 Decimal expansion of arctan(sqrt(Omega)), where Omega=LambertW(1) is the Omega constant.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A342360 Decimal expansion of 1/(Omega+1)^2, where Omega=LambertW(1) is the Omega constant.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula