cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A243446 Decimal expansion of 3/(2*sqrt(Pi)).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 05 2014

Keywords

Comments

Expectation of the maximum of a size 3 sample from a normal (0,1) distribution.

Examples

			0.846284375321634430422119177341158878766...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Links

Crossrefs

Cf. A087197, A188340.

Programs

  • Mathematica
    RealDigits[3/(2*Sqrt[Pi]), 10, 99] // First
  • PARI
    3/(2*sqrt(Pi)) \\ G. C. Greubel, Jan 09 2017

A243447 Decimal expansion of 1-9/(4*Pi)+sqrt(3)/(2*Pi), an extreme value constant.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 05 2014

Keywords

Comments

Variance of the maximum of a size 3 sample from a normal (0,1) distribution.

Examples

			0.55946720379736701379568631398017...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Links

Crossrefs

Cf. A087197, A188340, A243446.

Programs

  • Mathematica
    RealDigits[1 - 9/(4*Pi) + Sqrt[3]/(2*Pi), 10, 100] // First
  • PARI
    1-9/4/Pi + sqrt(3)/2/Pi \\ Charles R Greathouse IV, Sep 28 2022

A243448 Decimal expansion of 6*arcsec(sqrt(3))/Pi^(3/2), an extreme value constant.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 05 2014

Keywords

Comments

Expectation of the maximum of a size 4 sample from a normal (0,1) distribution.

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Crossrefs

Cf. A087197, A188340, A243446, A243447.

Programs

  • Mathematica
    RealDigits[6*ArcSec[Sqrt[3]]/Pi^(3/2), 10, 99] // First

A243452 Decimal expansion of the variance of the maximum of a size 4 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 05 2014

Keywords

Examples

			0.4917152368747417606817470099858870229...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Crossrefs

Cf. A087197, A188340, A243446, A243447, A243448.

Programs

  • Mathematica
    RealDigits[1 + Sqrt[3]/Pi - 36*ArcSec[Sqrt[3]]^2/Pi^3, 10, 102] // First

Formula

1 + sqrt(3)/Pi - 36*arcsec(sqrt(3))^2/Pi^3.

A243453 Decimal expansion of the expectation of the maximum of a size 5 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 05 2014

Keywords

Examples

			1.1629644736405196127722679885505014941...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Links

Crossrefs

Cf. A087197, A188340, A243446, A243447, A243448, A243452.

Programs

  • Mathematica
    RealDigits[5/Sqrt[Pi] - 15*ArcCsc[Sqrt[3]]/Pi^(3/2), 10, 102] // First
  • PARI
    5/sqrt(Pi) - 15*asin(1/sqrt(3))/Pi^(3/2) \\ G. C. Greubel, Feb 01 2017

Formula

5/sqrt(Pi) - 15*arccsc(sqrt(3))/Pi^(3/2).

A258146 Decimal expansion of (1 - 2/Pi)/2: ratio of the area of a circular segment with central angle Pi/2 and the area of the corresponding circular half-disk.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, May 31 2015

Keywords

Comments

The formula for the ratio of the area of a circular segment with central angle alpha and the area of one half of the corresponding circular disk is (alpha - sin(alpha))/Pi. Here alpha = Pi/2.
This is also the ratio of the area of a circular disk without a central inscribed rectangle (2*x, 2*y) together with the two opposite circular segments each with central angle beta and the area of the circular disk. This is the analog of the ratio of the volume of a sphere with missing central cylinder symmetric hole of length 2*y and the area of the sphere. See a comment on A019699. In two dimensions this problem is not remarkable, because the radius R of the circle does matter. The formula is here: area ratio ar = 1 - (beta + sin(beta)/Pi) where beta = arcsin(2*yhat*sqrt(1-yhat^2)), with yhat = y/R, and beta = Pi - alpha from above.
The astonishing result from three dimensions, ar_3 = yhat^3, could suggest ar = yhat^2, which is wrong. Thanks to Sven Heinemeyer for inspiring me to look into this.
Essentially the same digit sequence as A188340. - R. J. Mathar, Jun 12 2015

Links

Crossrefs

Cf. A019699, A060294, A258147, A258148.

Programs

Formula

Area ratio ar = (1 - 2/Pi)/2 = 0.181690113816209...
For Buffon's constant 2/Pi see A060294.

A243454 Decimal expansion of the variance of the maximum of a size 5 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 05 2014

Keywords

Examples

			0.44753406902066198876568465773...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Crossrefs

Cf. A087197, A188340, A243446, A243447, A243448, A243452, A243453.

Programs

  • Mathematica
    RealDigits[1 - 25/Pi + 150*ArcCsc[Sqrt[3]]/Pi^2 + 5*Sqrt[3]*ArcSec[2*Sqrt[2/3]]/Pi^2 - 225*ArcCsc[Sqrt[3]]^2/Pi^3, 10, 101] // First

Formula

1 - 25/Pi + 150*arccsc(sqrt(3))/Pi^2 + 5*sqrt(3)*arcsec(2*sqrt(2/3))/Pi^2 - 225*arccsc(sqrt(3))^2/Pi^3.

A243525 Decimal expansion of the variance of the maximum of a size 6 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 06 2014

Keywords

Examples

			0.415927108983248119181409058601893424...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Crossrefs

Cf. A188340, A243447, A243452, A243454, A243523.

Programs

  • Mathematica
    digits = 100; v[6] = -((225*(Pi*(Pi - 4*ArcCsc[Sqrt[3]]) + 2*NIntegrate[ArcSin[Sqrt[3]*Sqrt[1/(8 - Tan[x]^2)]], {x, 0, ArcCsc[Sqrt[3]]}, WorkingPrecision -> digits+5])^2)/(4*Pi^5)) + (5*Sqrt[3]*(Pi - 3*ArcCsc[2*Sqrt[2/3]]))/Pi^2 + 1; RealDigits[v[6], 10, digits] // First

Formula

-((225*(Pi*(Pi-4*arccsc(Sqrt(3))) + 2*integral_(x=0..arccsc(sqrt(3)))(arcsin(sqrt(3)*sqrt(1/(8-tan(x)^2)))))^2)/(4*Pi^5))+(5*sqrt(3)*(Pi-3*arccsc(2*sqrt(2/3))))/Pi^2+1

A243526 Decimal expansion of the variance of the maximum of a size 7 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 06 2014

Keywords

Examples

			0.39191777612675045281968496580009199872...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Crossrefs

Cf. A188340, A243447, A243452, A243454, A243525, A243524.

Programs

  • Mathematica
    digits = 99; v[7] = -((441*(Pi*(Pi - 5*ArcCsc[Sqrt[3]]) + 5*NIntegrate[ArcSin[Sqrt[3]*Sqrt[1/(8 - Tan[x]^2)]], {x, 0, ArcCsc[Sqrt[3]]},
WorkingPrecision -> digits + 5])^2)/(4*Pi^5)) + (35*Sqrt[3]*(Pi*(Pi - 4*ArcCsc[2*Sqrt[2/3]]) + 2*NIntegrate[ArcSin[Sqrt[6]*Sqrt[1/(15 - Tan[x]^2)]], {x, 0, ArcCsc[2*Sqrt[2/3]]}, WorkingPrecision -> digits + 5]))/(4*Pi^3) + 1; RealDigits[v[7], 10, digits] // First

Formula

-((441*(Pi*(Pi - 5*arccsc(sqrt(3))) + 5*integral_(x=0..arccsc(sqrt(3)) )(arcsin(sqrt(3)*sqrt(1/(8 - tan(x)^2))), {x, 0, arccsc(sqrt(3))} ))^2)/(4*Pi^5)) + (35*sqrt(3)*(Pi*(Pi - 4*arccsc(2*sqrt(2/3))) + 2*integral_(x=0..arccsc(2*sqrt(2/3)))(arcsin(sqrt(6)*sqrt(1/(15 - tan(x)^2))), {x, 0, arccsc(2*sqrt(2/3))} )))/(4*Pi^3) + 1

A243964 Decimal expansion of the variance of the maximum of a size 8 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 16 2014

Keywords

Comments

According to Steven Finch, no exact expression of this moment is known.

Examples

			0.3728971432867289942202112287621146...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Crossrefs

Cf. A188340 v(2), A243447 v(3), A243452 v(4), A243454 v(5), A243525 v(6), A243526 v(7), A243961 mu(8).

Programs

  • Mathematica
    digits = 101; m0 = 5; dm = 5; f[x_] := 1/ Sqrt[2*Pi]*Exp[-x^2/2]; F[x_] := 1/2*Erf[x/Sqrt[2]] + 1/2; Clear[mu8]; mu8[m_] := mu8[m] = 8*NIntegrate[x*F[x]^7*f[x], {x, -m , m}, WorkingPrecision -> digits+5, MaxRecursion -> 20]; mu8[m0]; mu8[m = m0+dm]; While[RealDigits[mu8[m]] != RealDigits[mu8[m-dm]], Print["m1 = ", m]; m = m+dm]; m8 = mu8[m]; Clear[v, m]; v[m_] := v[m] = 8*NIntegrate[x^2*F[x]^7*f[x], {x, -m , m}, WorkingPrecision -> digits+5, MaxRecursion -> 20]; v[m0]; v[m = m0+dm]; While[RealDigits[v[m]] != RealDigits[v[m-dm]], Print["m2 = ", m]; m = m+dm]; v8 = v[m]-m8^2; RealDigits[v8, 10, digits] // First

Formula

integral_(-infinity..infinity) 8*x^2*F(x)^7*f(x) dx - mu(8)^2, where f(x) is the normal (0,1) density and F(x) its cumulative distribution, mu(8) being the moment A243961.
Showing 1-10 of 10 results.

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