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.

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A272413 Asymptotic mean (normalized by n) of the second longest cycle in a random permutation on n symbols.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 29 2016

Keywords

Examples

			0.20958087428418581398902965781534955690113103201623433...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.

Links

Crossrefs

Cf. A084945, A247398.

Programs

  • Mathematica
    digits = 98; NIntegrate[1 - Exp[ExpIntegralEi[-x]]*(1 - ExpIntegralEi[-x]), {x, 0, 200}, WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First

Formula

Integral_{0..infinity} 1 - exp(Ei(-x))*(1 - Ei(-x)) dx, where Ei is the exponential integral.

A244261 Decimal expansion of c = 2.4149..., a random mapping statistics constant such that the asymptotic expectation of the maximum rho length (graph diameter) in a random n-mapping is c*sqrt(n).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 24 2014

Keywords

Examples

			2.4149010237176162411...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 288.
  • P. Flajolet and A. M. Odlyzko, Random Mapping Statistics, Advances in Cryptology - EUROCRYPT '89, J.-J. Quisquater and J. Vandewalle (eds.), Lecture Notes in Computer Science, Springer Verlag, 1990, pp. 329-354.

Crossrefs

Cf. A084945, A244067, A244258.

Programs

  • Maple
    evalf(sqrt(Pi/2)*Int(1 - exp(Ei(-x) - Int((exp(-y)/y)*(1 - exp(-2*(y/(exp(x - y) - 1)))), y=0..x)), x=0..infinity)); # Vaclav Kotesovec, Aug 12 2019
  • Mathematica
    digits = 20; m0 = 100; dm = 10; I0[x_?NumericQ] := NIntegrate[(Exp[-y]/y)*(1 - Exp[-2*(y/(Exp[x - y] - 1))]), {y, 0, x}, WorkingPrecision -> digits+5]; Clear[f]; f[m_] := f[m] = Sqrt[Pi/2]* NIntegrate[1 - Exp[ExpIntegralEi[-x] - I0[x]], {x, 0, m}, WorkingPrecision -> digits+5]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits+5] != RealDigits[f[m - dm], 10, digits+5], Print["m = ", m]; m = m + dm]; RealDigits[f[m], 10, digits] // First

Formula

I(x) = integral_(0..x) (exp(-y)/y)*(1 - exp(-2*(y/(exp(x - y) - 1)))) dy,
c = sqrt(Pi/2)*integral_(0..infinity) 1 - exp(Ei(-x) - I(x)) dx, where Ei is the exponential integral function.

A272414 Asymptotic variance (normalized by n^2) of the second longest cycle in a random permutation on n symbols.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 29 2016

Keywords

Examples

			0.012553790635905878147980035846601986785508301199365...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.

Links

Crossrefs

Cf. A084945, A247398, A272413.

Programs

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

Formula

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

A225337 Incrementally largest terms in the continued fraction of the Golomb-Dickman constant.

Original entry on oeis.org

0, 1, 22, 28, 43, 48, 66, 491, 1706, 4763, 38371
Offset: 1

Views

Author

Eric W. Weisstein, Jul 25 2013

Keywords

Links

Crossrefs

Cf. A225363 (positions of largest terms).
Cf. A225336 (continued fraction of the Golomb-Dickman constant).
Cf. A084945 (decimal expansion of the Golomb-Dickman constant).

A225363 Positions of incrementally largest terms in the continued fraction expansion of the Golomb-Dickman constant.

Original entry on oeis.org

0, 1, 6, 24, 39, 50, 52, 72, 259, 1002, 4610
Offset: 1

Views

Author

Eric W. Weisstein, Jul 25 2013

Keywords

Links

Crossrefs

Cf. A225337 (incrementally largest terms).
Cf. A225336 (continued fraction of the Golomb-Dickman constant).
Cf. A084945 (decimal expansion of the Golomb-Dickman constant).

A225364 Position of first occurrence of n in the continued fraction for the Golomb-Dickman constant.

Original entry on oeis.org

1, 8, 9, 30, 25, 18, 110, 242, 59, 100, 12, 71, 28, 153, 225, 114, 159, 66, 75, 102, 924, 6, 631, 150, 299, 434, 701, 24, 1687, 196, 1482, 779, 1552, 2658, 505, 496, 255, 46, 1626, 183, 2551, 1083, 39, 665, 1419, 678, 1676, 50, 1027, 2047, 3726, 1309, 2327
Offset: 1

Views

Author

Eric W. Weisstein, Jul 25 2013

Keywords

Examples

			The c.f. for lambda is [0; 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, ..], so
a(1) = 1 (1 occurs first at term a_1).
a(2) = 8 (2 occurs first at term a_8).
a(3) = 9 (3 occurs first at term a_9).

Links

Crossrefs

Cf. A225336 (continued fraction of the Golomb-Dickman constant).
Cf. A084945 (decimal expansion of the Golomb-Dickman constant).

A229195 Beginning position of n in the decimal expansion of the Golomb-Dickman constant.

Original entry on oeis.org

15, 28, 2, 4, 3, 10, 1, 17, 8, 6, 28, 76, 203, 77, 103, 120, 206, 37, 46, 60, 44, 204, 256, 197, 2, 79, 42, 88, 52, 5, 272, 27, 4, 586, 405, 12, 23, 32, 25, 95, 104, 36, 41, 3, 35, 288, 82, 146, 191, 64, 14, 59
Offset: 0

Views

Author

Eric W. Weisstein, Sep 15 2013

Keywords

Links

Crossrefs

Cf. A084945 (decimal expansion of the Golomb-Dickman constant).

A241033 Decimal expansion of 'c', a constant such that in N steps, the mean longest random walk duration records grow as c*N.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 15 2014

Keywords

Examples

			0.626507598767175449080634422596159044653245248...

Links

Crossrefs

Cf. A084945 (the analog constant for i.i.d. random variables).

Programs

  • Mathematica
    2*NIntegrate[Log[1 + 1/(2*Sqrt[Pi])*Gamma[-1/2, x]], {x, 0, Infinity}, WorkingPrecision -> 105] // RealDigits[#, 10, 100]& // First

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]]

A272415 Asymptotic mean (normalized by n) of the third longest cycle in a random permutation on n symbols.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 29 2016

Keywords

Examples

			0.0883160988831536310105425664298767011723643204511633304667874093...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.

Links

Crossrefs

Cf. A084945, A247398, A272413, A272414.

Programs

  • Mathematica
    digits = 98; NIntegrate[1 - E^ExpIntegralEi[-x]*(1 - ExpIntegralEi[-x] + (1/2)*ExpIntegralEi[-x]^2), {x, 0, 100}, WorkingPrecision -> digits + 5] // Join[{0}, RealDigits[#, 10, digits][[1]]]&

Formula

Integral_{0..infinity} 1 - e^Ei(-x)*(1 - Ei(-x) + (1/2)*Ei(-x)^2) dx, where Ei is the exponential integral.
Previous Showing 11-20 of 25 results. Next

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