A242816 -id:A242816 - cp's OEIS frontend

A086231 Decimal expansion of value of Watson's integral.

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

Offset: 1

Eric W. Weisstein, Jul 12 2003

			1.51638605915197801815601215968142077995538704445226267656698...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 322.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 40.
M. Lawrence Glasser and I. John Zucker, Extended Watson integrals for the cubic lattices, Proceedings of the National Academy of Sciences, Vol. 74, No. 5 (1977), pp. 1800-1801, alternative link.
Anthony J. Guttmann, Lattice Green's functions in all dimensions, J. Phys. A.: Math. Theor., Vol. 43, No. 30 (2010) 305205.
George N. Watson, Three triple integrals, The Quarterly Journal of Mathematics, Vol. os-10, No. 1 (1939), pp. 266-276.
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Cf. A002896, A086230, A242812, A242813, A242814, A242815, A242816, A273086.

C := ComplexField(); (Sqrt(3)-1)*(Gamma(1/24)*Gamma(11/24))^2/(32*Pi(C)^3); // G. C. Greubel, Jan 07 2018

Maple

evalf((sqrt(3)-1)*(GAMMA(1/24)*GAMMA(11/24))^2 / (32*Pi^3),120); # Vaclav Kotesovec, Sep 16 2014

Mathematica

RealDigits[ N[ Sqrt[6]/32/Pi^3*Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24], 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)

PARI

(sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2 / (32*Pi^3) \\ Altug Alkan, Apr 13 2016

Formula

Equals (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2/(32*Pi^3). - G. C. Greubel, Jan 07 2018

Equals 1/(1 - A086230). - Amiram Eldar, Aug 28 2020

Equals Sum_{k>=0} A002896(k)/36^k. - Vaclav Kotesovec, Apr 23 2023

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

Eric W. Weisstein, Jul 12 2003

			0.0729126499593839984697453553883...

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

Marc Mezzarobba, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

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

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

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

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

Jean-François Alcover, May 23 2014

			1.1563081248...

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

Marc Mezzarobba, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

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

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

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

intnumosc(t=0,exp(-t)*besseli(0,t/5)^5,Pi*5) \\ Charles R Greathouse IV, Oct 23 2023

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

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

Jean-François Alcover, May 23 2014

			1.09390631558784799668327...

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

Marc Mezzarobba, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

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

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

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

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

More terms from Alois P. Heinz, May 23 2014

A242761 Decimal expansion of the escape probability for a random walk on the 3-D cubic lattice (a Polya random walk constant).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 22 2014

			0.6594626704490008571737268155670971...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9, p. 322.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Polya's Random Walk Constants

Cf. A086230, A086231, A086232-A086236, A043546, A293237, A293238, A242812-A242816.

SetDefaultRealField(RealField(100)); R:= RealField(); (16*Sqrt(2/3)*Pi(R)^3)/(Gamma(1/24)*Gamma(5/24)*Gamma(7/24)*Gamma(11/24)); // G. C. Greubel, Oct 26 2018

p = (16*Sqrt[2/3]*Pi^3)/(Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]); RealDigits[p, 10, 100] // First

default(realprecision, 100); (16*sqrt(2/3)*Pi^3)/(gamma(1/24)* gamma(5/24)*gamma(7/24)*gamma(11/24)) \\ G. C. Greubel, Oct 26 2018

Equals (16*sqrt(2/3)*Pi^3)/(Gamma(1/24)*Gamma(5/24)*Gamma(7/24)*Gamma(11/24)), where Gamma is the Euler Gamma function.

cp's OEIS Frontend

A086231 Decimal expansion of value of Watson's integral.

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

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

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

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Maple

Mathematica

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A242761 Decimal expansion of the escape probability for a random walk on the 3-D cubic lattice (a Polya random walk constant).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula