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.

A271948 Decimal expansion of a constant related to the variance of the number of vertices of the largest tree associated with a random mapping on n symbols.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 20 2016

Keywords

Examples

			0.049469852279228075333485464056253836603725107670028013295315781039...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 289.

Links

Crossrefs

Cf. A084945, A143297, A244067, A244258, A244261, A261873, A271871.

Programs

  • Mathematica
    digits = 96; F[x_] := 1 - Exp[-x]/Sqrt[Pi*x] - Erf[Sqrt[x]]; Clear[f, g];
f[m_] := f[m] = 2 NIntegrate[(1 - (1 - F[x])^-1), {x, 0, m}, WorkingPrecision -> digits + 10]; f[m = 100]; f[m = 2 m]; Print["m = ", m]; While[RealDigits[f[m], 10, digits + 5][[1]] != RealDigits[f[m/2], 10, digits + 5][[1]], m = 2 m; Print["m = ", m]];
g[m_] := g[m] = (8/3) NIntegrate[(1 - (1 - F[x])^-1)*x, {x, 0, m}, WorkingPrecision -> digits + 10]; g[m = 100]; g[m = 2 m]; Print["m = ", m]; While[RealDigits[g[m], 10, digits + 5][[1]] != RealDigits[g[m/2], 10, digits + 5][[1]], m = 2 m; Print["m = ", m]];
Join[{0}, RealDigits[g[m] - f[m]^2, 10, digits][[1]]]

A261873 Decimal expansion of H(1/2,1), a constant appearing in the asymptotic variance of the largest component of random mappings on n symbols, expressed as H(1/2,1)*n^2.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 04 2015

Keywords

Examples

			0.037007216582290303209923789448919330070073980621328473638505730597...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 289.

Crossrefs

Cf. A143297.

Programs

  • Mathematica
    digits = 105; h1 = (8/3)*NIntegrate[(1 - Exp[ExpIntegralEi[-x]/2])*x, {x, 0, Infinity}, WorkingPrecision -> digits + 10]; h2 = 4*NIntegrate[1 - Exp[ExpIntegralEi[-x]/2], {x, 0, Infinity}, WorkingPrecision -> digits + 10]^2 ; Join[{0}, RealDigits[h1 - h2, 10, digits] // First]

Formula

H(1/2,1) = (8/3) Integral_{0..infinity} (1-exp(Ei(-x)/2)) x dx - A143297^2, where A143297 is G(1/2,1), using Finch's notation.

A245422 Decimal expansion of the coefficient c appearing in the expression of the asymptotic expected shortest cycle in a random n-cyclation as c*sqrt(n).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 08 2014

Keywords

Examples

			1.457270879273653853694454068120047059660530020235224659213297...

Links

Crossrefs

Cf. A143297 (analog in the case of the expected *longest* cycle in a random cyclation).

Programs

  • Mathematica
    digits = 102; (Sqrt[Pi]/2)*NIntegrate[Exp[-x - ExpIntegralEi[-x]/2], {x, 0, Infinity}, WorkingPrecision -> digits+10] // RealDigits[#, 10, digits]& // First

Formula

(sqrt(Pi)/2)*integral_{0..infinity} exp(-x - Ei(-x)/2), where Ei is the exponential integral function.

A271871 Decimal expansion of a constant related to the expected number of vertices of the largest tree associated with a random mapping on n symbols.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 20 2016

Keywords

Examples

			0.48349834715442550092402636085075619444924667954133810432926494155247...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 289.

Links

Crossrefs

Cf. A084945, A143297, A244067, A244258, A244261, A261873.

Programs

  • Mathematica
    digits = 98; F[x_] := 1 - Exp[-x]/Sqrt[Pi*x] - Erf[Sqrt[x]]; Clear[f]; f[m_] := f[m] = 2 NIntegrate[1-(1-F[x])^-1, {x, 0, m}, WorkingPrecision -> digits+10]; f[m = 100]; f[m = 2 m]; Print["m = ", m]; While[ RealDigits[ f[m], 10, digits + 5][[1]] != RealDigits[f[m/2], 10, digits + 5][[1]], m = 2 m; Print["m = ", m]]; RealDigits[f[m/2], 10, digits + 5][[1]]

A272429 Asymptotic mean (normalized by n) of the second largest connected component in a random mapping on n symbols.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 29 2016

Keywords

Examples

			0.17090961985966239214460728413311738704719072962628832355853881...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random mapping statistics, p. 290.

Links

Crossrefs

Cf. A084945, A143297, A261873.

Programs

  • Mathematica
    digits = 95; Ei = ExpIntegralEi; 2*NIntegrate[1 - E^(Ei[-x]/2)*(1 - Ei[-x]/2), {x, 0, 200}, WorkingPrecision -> digits + 5] // RealDigits[#, 10, digits]& // First

Formula

2*integral_{0..infinity} 1 - e^(Ei(-x)/2)*(1 - Ei(-x)/2) dx, where Ei is the exponential integral.

A272430 Asymptotic variance (normalized by n^2) of the second largest connected component in a random mapping on n symbols.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 29 2016

Keywords

Examples

			0.01862022330678138872140657036234315043193560144957499823184259199928...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random mapping statistics, p. 290.

Links

Crossrefs

Cf. A084945, A143297, A261873, A272429.

Programs

  • Mathematica
    digits = 98; Ei = ExpIntegralEi; (8/3)*NIntegrate[x*(1 - E^(Ei[-x]/2)*(1 - Ei[-x]/2)), {x, 0, 200}, WorkingPrecision -> digits + 5] - 4*NIntegrate[1 - E^(Ei[-x]/2)*(1 - Ei[-x]/2), {x, 0, 200}, WorkingPrecision -> digits + 5]^2 // Join[{0}, RealDigits[#, 10, digits][[1]]]&

Formula

(8/3)*integral_{0..infinity} x*(1 - e^(Ei(-x)/2)*(1 - Ei(-x)/2)) dx - 4*(integral_{0..infinity} 1 - e^(Ei(-x)/2)*(1 - Ei(-x)/2) dx)^2, where Ei is the exponential integral.
Showing 1-6 of 6 results.

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