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 1-6 of 6 results.

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).

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.

A243523 Decimal expansion of the expectation of the maximum of a size 6 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 06 2014

Keywords

Examples

			1.26720636061147129764934888186399444269365...

References

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

Crossrefs

Cf. A087197, A243446, A243448, A243453.

Programs

  • Mathematica
    digits=100; mu[6] = (15*(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* Pi^(5/2)) ; RealDigits[mu[6], 10, digits] // First

Formula

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

A243524 Decimal expansion of the expectation of the maximum of a size 7 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 06 2014

Keywords

Examples

			1.3521783756069043992289221668285773452932858435...

References

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

Crossrefs

Cf. A087197, A243446, A243448, A243453, A243523.

Programs

  • Mathematica
    digits = 100; mu[7] = (21*(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* Pi^(5/2)); RealDigits[mu[7], 10, digits] // First

Formula

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

A243961 Decimal expansion of the expectation of the maximum of a size 8 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 16 2014

Keywords

Comments

According to Steven Finch, no exact expression of this moment mu(8) is known, unlike the moments mu(n) for n<8.

Examples

			1.423600306045277753078324649306257253...

References

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

Crossrefs

Cf. A087197 mu(2), A243446 mu(3), A243448 mu(4), A243453 mu(5), A243523 mu(6), A243524 mu(7).

Programs

  • Mathematica
    digits = 100; 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["m = ", m]; m = m + dm]; RealDigits[mu8[m], 10, digits] // First

Formula

integral_(-infinity..infinity) 8*x*F(x)^7*f(x) dx, where f(x) is the normal (0,1) density and F(x) its cumulative distribution.
Showing 1-6 of 6 results.

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