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 1-4 of 4 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]]]

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 1-4 of 4 results.

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