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 1-7 of 7 results.

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).

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 1-7 of 7 results.

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