A342360 -id:A342360 - 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))).

A342361 Decimal expansion of 1/(omega+1)^2, where omega=1/LambertW(1).

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 09 2021

			0.1309689005663456008580754336956370484226429615564731843

Cf. A342359, A342360, A030178, A030797.

Omega=LambertW[1]; xi=ArcTan[Sqrt[Omega]]; N[Sin[xi]^4,120]
omega=1/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]
RealDigits[1/(1/LambertW[1]+1)^2,10,120][[1]] (* Harvey P. Dale, Mar 26 2025 *)

my(Omega=lambertw(1), xi=atan(sqrt(Omega))); sin(xi)^4

PARI
```
1/(1/lambertw(1)+1)^2
```

Equals Integral_{t=0..1} (-t/W(-1,-t*Omega^omega))^omega, where omega = 1/Omega = 1/LambertW(1).

Equals sin(A342359)^4 = 1/(A030797+1)^2 = (1-sqrt(A342360))^2.

cp's OEIS Frontend

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

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A342361 Decimal expansion of 1/(omega+1)^2, where omega=1/LambertW(1).

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