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 1-10 of 10 results.

A086230 Decimal expansion of probability that a random walk on a 3-D lattice returns to the origin.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jul 12 2003

Keywords

Comments

Pólya (1921) proved that this constant is < 1. McCrea and Whipple (1940) evaluated it by 0.34. - Amiram Eldar, Aug 28 2020

Examples

			0.340537329550999142826273184432902896710608217124302097763236105377791969...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 322-331.

Links

Crossrefs

Cf. A086231, A086232, A086233, A086234, A086235, A086236.

Programs

  • Magma
    C := ComplexField(); 1 - (16*Sqrt(2/3)*Pi(C)^3)/(Gamma(1/24)* Gamma(5/24)*Gamma(7/24)*Gamma(11/24)); // G. C. Greubel, Jan 25 2018
  • Mathematica
    RealDigits[1 - (16*Sqrt[2/3]*Pi^3) / (Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]), 10, 102] // First (* Jean-François Alcover, Feb 08 2013, after Eric W. Weisstein *)
  • PARI
    1-32*Pi^3/sqrt(6)/gamma(1/24)/gamma(5/24)/gamma(7/24)/gamma(11/24) \\ Charles R Greathouse IV, Jul 22 2013

Formula

Equals 1 - (16*Sqrt(2/3)*Pi^3)/(Gamma(1/24)* Gamma(5/24)*Gamma(7/24)* Gamma(11/24)). - G. C. Greubel, Jan 25 2018
Equals 1 - 1/A086231. - Amiram Eldar, Aug 28 2020

A086232 Decimal expansion of probability that a random walk on a 4-d lattice returns to the origin.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jul 12 2003

Keywords

Examples

			0.19320167322498393734187097332936916057587338645...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 323.

Links

Crossrefs

Cf. A086230, A086233, A086234, A086235, A086236, A242812.

Programs

  • Mathematica
    First[RealDigits[1 - 1/NIntegrate[BesselI[0, t/4]^4 * Exp[ -t], {t, 0, Infinity}, PrecisionGoal->50, WorkingPrecision->350]]] (* Ryan Propper, Jul 12 2005 *)

Formula

Equals 1 - 1/A242812. - Amiram Eldar, Aug 28 2020

Extensions

More terms from Ryan Propper, Jul 12 2005
a(51) corrected and more terms using the data at A242812 added by Amiram Eldar, Aug 28 2020

A086236 Decimal expansion of probability that a random walk on an 8-d lattice returns to the origin.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jul 12 2003

Keywords

Examples

			0.0729126499593839984697453553883...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 323.

Links

Crossrefs

Cf. A086230, A086232, A086233, A086234, A086235, A242816.

Formula

Equals 1 - 1/A242816. - Amiram Eldar, Aug 28 2020

Extensions

More terms using the data at A242816 added by Amiram Eldar, Aug 28 2020

A086233 Decimal expansion of the probability that a random walk on the 5-d simple cubic (hypercubic) lattice returns to the origin.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jul 12 2003

Keywords

Examples

			0.1351786098206552...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 323.

Links

Crossrefs

Cf. A086230, A086232, A086234, A086235, A086236, A242813.

Formula

Equals 1-1/A242813. - Andrey Zabolotskiy, Dec 28 2018

Extensions

More terms from Andrey Zabolotskiy, Dec 28 2018 based on A242813

A086235 Decimal expansion of probability that a random walk on a 7-d lattice returns to the origin.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jul 12 2003

Keywords

Examples

			0.08584493411337900918810813478517355664...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 323.

Links

Crossrefs

Cf. A086230, A086232, A086233, A086234, A086236, A242815.

Formula

Equals 1 - 1/A242815. - Amiram Eldar, Aug 28 2020

Extensions

More terms using the data at A242815 added by Amiram Eldar, Aug 28 2020

A242812 Decimal expansion of the expected number of returns to the origin of a random walk on a 4-d lattice.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 23 2014

Keywords

Examples

			1.239467121848481712678697664859...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Links

Crossrefs

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242813.

Programs

  • Maple
    m4:= int(exp(-t)*BesselI(0, t/4)^4, t=0..infinity):
s:= convert(evalf(m4, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014
  • Mathematica
    digits = 50; NIntegrate[BesselI[0, t/4]^4*Exp[-t], {t, 0, Infinity}, PrecisionGoal -> digits, WorkingPrecision -> 350] // RealDigits [#, 10, digits]& // First (* after Ryan Propper *)

Formula

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = Integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086232). - Amiram Eldar, Aug 28 2020

Extensions

More terms from Alois P. Heinz, May 23 2014

A242813 Decimal expansion of the expected number of returns to the origin of a random walk on a 5-d lattice.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 23 2014

Keywords

Examples

			1.1563081248...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Links

Crossrefs

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242814, A242815, A242816.

Programs

  • Maple
    m5:= int(exp(-t)*BesselI(0, t/5)^5, t=0..infinity):
s:= convert(evalf(m5, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014
  • Mathematica
    d = 5; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 10]& // First
  • PARI
    intnumosc(t=0,exp(-t)*besseli(0,t/5)^5,Pi*5) \\ Charles R Greathouse IV, Oct 23 2023

Formula

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086233). - Amiram Eldar, Aug 28 2020

Extensions

More terms from Alois P. Heinz, May 23 2014

A242814 Decimal expansion of the expected number of returns to the origin of a random walk on a 6-d lattice.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 23 2014

Keywords

Examples

			1.1169633732...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Links

Crossrefs

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813.

Programs

  • Maple
    m6:= int(exp(-t)*BesselI(0, t/6)^6, t=0..infinity):
s:= convert(evalf(m6, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014
  • Mathematica
    d = 6; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 8]& // First
  • PARI
    intnumosc(t=0,exp(-t)*besseli(0,t/6)^6,12*Pi) \\ Charles R Greathouse IV, Oct 23 2023

Formula

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086234). - Amiram Eldar, Aug 28 2020

Extensions

More terms from Alois P. Heinz, May 23 2014

A242816 Decimal expansion of the expected number of returns to the origin of a random walk on an 8-d lattice.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 23 2014

Keywords

Examples

			1.0786470120...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Links

Crossrefs

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813, A242814, A242815.

Programs

  • Maple
    m8:= int(exp(-t)*BesselI(0, t/8)^8, t=0..infinity):
s:= convert(evalf(m8, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014
  • Mathematica
    d = 8; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 7]& // First

Formula

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = Integral_{t>0} exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086236). - Amiram Eldar, Aug 28 2020

Extensions

More terms from Alois P. Heinz, May 23 2014

A242815 Decimal expansion of the expected number of returns to the origin of a random walk on a 7-d lattice.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 23 2014

Keywords

Examples

			1.09390631558784799668327...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Links

Crossrefs

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813, A242814, A242816.

Programs

  • Maple
    m7:= int(exp(-t)*BesselI(0, t/7)^7, t=0..infinity):
s:= convert(evalf(m7, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014
  • Mathematica
    d = 7; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 7]& // First

Formula

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = Integral_{t>0} exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086235). - Amiram Eldar, Aug 28 2020

Extensions

More terms from Alois P. Heinz, May 23 2014
Showing 1-10 of 10 results.

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