Gleb Koloskov - cp's OEIS frontend

A346962 Decimal expansion of Integral_{x=-1/e..0} LambertW(x)/LambertW(-1,x) dx.

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

Offset: 0

Gleb Koloskov, Aug 09 2021

			0.0556295897289868954612901466941050684561294869117252169349398695712429...

Cf. A000272, A000312, A068985, A346963.

evalf(Integrate(LambertW(x)/LambertW(-1, x), x = -exp(-1)..0), 120); # Vaclav Kotesovec, Aug 23 2021

N[Integrate[LambertW[x]/LambertW[-1,x],{x,-1/E,0}],120]

exp(-1)+intnum(x=0,1,log(x)*x^(1/(1-x))/(1-x))

Equals (1/e) + Integral_{x=0..1} log(x)*x^(1/(1-x))/(1-x) dx.

Equals (1/e) - Sum_{n>0} n^(n-2)/(n+1)^(n+1) = A068985-Sum_{n>0} A000272(n)/A000312(n+1).

A346963 Decimal expansion of Integral_{x=-1/e..0} LambertW(x)*LambertW(-1,x) dx.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Aug 09 2021

			0.216577770436007277359024920060738331698735582255355693272331441694...

Cf. A000169, A007778, A068985, A135003, A135011, A346962.

evalf(Integrate(LambertW(x)*LambertW(-1, x), x = -exp(-1)..0), 120); # Vaclav Kotesovec, Aug 23 2021

N[Integrate[LambertW[x]*LambertW[-1,x],{x,-1/E,0}],120]

11*exp(-1)-4+sumpos(n=1,(1/(1+1./n))^n/(n*(n+1)^2))

Equals Integral_{x=-1/e..0} LambertW(x)*LambertW(-1,x) dx.
Equals (3/e) - 1 + Sum_{n>0} (n^(n-1)/(n+1)^(n+2))*(Gamma(n+2,n+1)/Gamma(n+2)).
Equals (11/e)-4+Sum_{n>0} n^(n-1)/(n+1)^(n+2) = A135011-4+Sum_{n>0} A000169(n)/A007778(n+1).

A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Jul 10 2021

			0.2288989481961786412366361253722...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A072334, A073742, A073747.

SetDefaultRealField(RealField(135)); 2/(Exp(2)-1)*Exp(2/(1-Exp(2))); // G. C. Greubel, Jun 11 2024

x/.FindRoot[LambertW[-x]-LambertW[-1,-x]==2, {x, 0.1, 0.3}, WorkingPrecision -> 110]
RealDigits[2/(E^2-1)*Exp[2/(1-E^2)], 10, 135][[1]] (* G. C. Greubel, Jun 11 2024 *)

PARI
```
exp(-cotanh(1))/sinh(1)
```

numerical_approx(2/(e^2-1)*exp(2/(1-e^2)), digits=135) # G. C. Greubel, Jun 11 2024

Equals exp(-coth(1))/sinh(1) = exp(-A073747)/A073742.
Equals (coth(1)-1)*exp(1-coth(1)) = (A073747-1)*exp(1-A073747).
Equals (coth(1)+1)/exp(1+coth(1)) = (A073747+1)/exp(1+A073747).
Equals 2/(e^2-1)*exp(2/(1-e^2)) = 2/(A072334^2-1)*exp(2/(1-A072334^2)).

A346062 Decimal expansion of the minimum value of the area of a rhombus circumscribed around a cosine-shaped lens, whose vertices lie on coordinate axes.

Original entry on oeis.org

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

Offset: 1

Gleb Koloskov, Jul 03 2021

Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and a rhombus circumscribed around S, whose vertices lie on coordinate axes.
This constant represents the value of the minimum area of such a rhombus KLMN with vertices K(0,2v), L(-2u,0), M(0,-2v), N(2u,0).
The rhombus touches S at the midpoints of its sides, A(u,v), B(-u,v), C(-u,-v), D(u,-v) which define a rectangle ABCD of the maximum area, inscribed in S, whose sides are parallel to coordinate axes. The constant u can be found as a root of equation x=cot(x) and is known as A069855, and v=cos(u)=u/sqrt(1+u^2).

			4.4887707055283605403232300252898136708822792436449257365...

Gleb Koloskov, Geometric illustration

Cf. A069855.

N[Minimize[{2 (x+Cot[x])^2 Sin[x],{x>0,x

u=solve(x=0.5,1,x-cotan(x));8*u^2/sqrt(1+u^2)

Equals 8*A069855^2/sqrt(1+A069855^2).

A345644 Decimal expansion of the radius of the circle tangent to the curves y=cos(x), y=-cos(x) and to the y-axis for x in [0,Pi/2].

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Jun 21 2021

Let r and (x,y) denote the radius of the circle and the point of tangency in the first quadrant, respectively.
Then r in [0,1] is the root of equation cos(r+sqrt(r^2-1+sqrt(1-r^2)))^2 = 1-sqrt(1-r^2),
r = 0.642707872546532445779211778468607918285047824...,
x = r+sqrt(r^2-1+sqrt(1-r^2)) = 1.066010072972971718857583783392083793389510385...,
y = sqrt(1-sqrt(1-r^2)) = 0.483620364074368181073730094271148302685427120...

			0.642707872546532445779211778468607918285047824...

Cf. A197016, A197017, A197018, A197019, A197020.

r = r /. FindRoot[Cos[r + Sqrt[-1 + r^2 + Sqrt[1 - r^2]]]^2 == 1 - Sqrt[1 - r^2], {r, 1/2}]; Show[Plot[Cos[x], {x, 0, Pi/2}], Plot[-Cos[x], {x, 0, Pi/2}], Graphics[Circle[{r, 0}, r]], PlotRange -> All, AspectRatio -> Automatic] (* Vaclav Kotesovec, Jul 01 2021 *)

solve(r=0,1,cos(r+sqrt(r^2-1+sqrt(1-r^2)))^2-1+sqrt(1-r^2))

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.

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

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

Original entry on oeis.org

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))).

A342209 Decimal expansion of logarithmic mean of Pi and e.

Original entry on oeis.org

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

Offset: 1

Gleb Koloskov, Mar 05 2021

The logarithmic mean of Pi and e lies between the arithmetic and geometric means of Pi and e.

			2.9248335452376948992733592...

B. C. Carlson, The logarithmic mean, The American Mathematical Monthly, 79(6) 1972, pp. 615-618.
Wikipedia, Logarithmic mean.

Cf. A000796 (Pi), A001113 (e), A019645, A074916, A074948, A074950, A074957.

RealDigits[(Pi - E)/(Log[Pi] - 1), 10, 100][[1]] (* Amiram Eldar, Mar 06 2021 *)

PARI
```
(Pi-exp(1))/(log(Pi)-1)
```

Equals (Pi-e)/(log(Pi)-1).

A341859 Decimal expansion of 4 - (8/5)*sqrt(5).

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 07 2021

In a triangle inscribed in a unit circle this is the maximal value of its inradius, such that a minimal closed Steiner chain of circles (10 circles) can be sandwiched between the incircle and circumcircle of the triangle.
It can be found as follows.
The squared distance between the centers of the two chain-defining circles is known to be d^2 = (R-r)^2 - 4*r*R*tan(Pi/n)^2.
On the other hand, the squared distance between the circumcenter and the incenter of triangle is known to be d^2 = R*(R-2*r).
Thus, in order to make a valid closed chain of circles, the inradius of triangle inscribed in the unit circle must be equal to 4*tan(Pi/n)^2.
Given that the maximum of such inradius is 0.5, the minimal number of chained circles is n=10, which gives the maximal value r = 4*tan(Pi/10)^2 = 0.42... < 0.5.

			0.4222912360003364857453221300299580232950106246215588411665644073...

Liang-Shin Hahn. Complex Numbers and Geometry (Mathematical Association of America Textbooks). The Mathematical Association of America, 1994, 140-141.

Wikipedia, Steiner Chain.

Cf. A019916, A001622.

RealDigits[4*Tan[18 Degree]^2, 10, 120][[1]]

PARI
```
4-8/5*sqrt(5)
```

Equals 4*A019916^2 = 4*tan(Pi/10)^2 = 4 - (8/5)*sqrt(5) = (4/5)*(7 - 4*phi) = (4/5)*(7 - 4*A001622), where phi is the golden ratio from A001622.

cp's OEIS Frontend

User: Gleb Koloskov

A346962 Decimal expansion of Integral_{x=-1/e..0} LambertW(x)/LambertW(-1,x) dx.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A346963 Decimal expansion of Integral_{x=-1/e..0} LambertW(x)*LambertW(-1,x) dx.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A346062 Decimal expansion of the minimum value of the area of a rhombus circumscribed around a cosine-shaped lens, whose vertices lie on coordinate axes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A345644 Decimal expansion of the radius of the circle tangent to the curves y=cos(x), y=-cos(x) and to the y-axis for x in [0,Pi/2].

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

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