A086231 -id:A086231 - cp's OEIS frontend

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

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.

A245067 Number of three-dimensional random walks with 2n steps in the wedge region x >= y >= z, beginning and ending at the origin without crossing the wedge boundary.

Original entry on oeis.org

1, 2, 12, 120, 1610, 25956, 474012, 9475752, 202921290, 4587734580, 108376022040, 2654745191280, 67043341981980, 1737717447946200, 46062204663294000, 1245096242017227360, 34239776369652506970, 956050033694583839220

Offset: 0

Jean-François Alcover, Nov 12 2014

			For 2n=4, the 12 acceptable walks are:
(0, 0, -1), (0, -1, -1), (0, 0, -1), (0 ,0, 0);
(0, 0, -1), (0, 0, 0), (0, 0, -1), (0 ,0, 0);
(0, 0, -1), (0, 0, 0), (1, 0, 0), (0 ,0, 0);
(0, 0, -1), (1, 0, -1), (0, 0, -1), (0 ,0, 0);
(0, 0, -1), (1, 0, -1), (1, 0, 0), (0 ,0, 0);
(1, 0, 0), (0, -1, -1), (0, 0, -1), (0 ,0, 0);
(1, 0, 0), (0, 0, 0), (0, 0, -1), (0 ,0, 0);
(1, 0, 0), (0, 0, 0), (1, 0, 0), (0 ,0, 0);
(1, 0, 0), (1, 0, -1), (0, 0, -1), (0 ,0, 0);
(1, 0, 0), (1, 0, -1), (1, 0, 0), (0 ,0, 0);
(1, 0, 0), (1, 1, 0), (0, 0, -1), (0 ,0, 0);
(1, 0, 0), (1, 1, 0), (1, 0, 0), (0 ,0, 0).

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 326.

Eric Weisstein's MathWorld, Polya's Random Walk Constants

Cf. A000108, A002896, A086230, A086231.

a[n_] := CatalanNumber[n]*HypergeometricPFQ[{1/2, -n-1, -n}, {2, 2}, 4]; Table[a[n], {n, 0, 20}]

a(n) = sum_{k=0..n} (2n)!*(2k)!/((n-k)!*(n+1-k)!*k!^2*(k+1)!^2).
a(n) = C(n) * 3F2(1/2, -n-1, -n; 2, 2; 4) where C(n) is the n-th Catalan number and 3F2 the hypergeometric function.
a(n) ~ 2^(2*n-4) * 3^(2*n+9/2) / (Pi^(3/2) * n^(9/2)). - Vaclav Kotesovec, Nov 13 2014
Recurrence: n*(n+2)^2*a(n) = 2*(2*n-1)*(10*n^2 + 2*n - 3)*a(n-1) - 36*(n-1)*(2*n-3)*(2*n-1)*a(n-2). - Vaclav Kotesovec, May 14 2016

A241624 Decimal expansion of 'c', one of Polya's Random Walk constants, related to the asymptotics of the number of 3-D random walks starting from and returning to the origin.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 19 2014

			0.53923817508158141965602919129297773...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.9 p. 325.

Cf. A086089, A086231.

m3 = A086231 = Sqrt[6]/32/Pi^3*Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]; c = 9/32*(m3 + 6/(Pi^2*m3)); RealDigits[c, 10, 100] // First

c = 9/32*(m3 + 6/(Pi^2*m3)), where m3 = A086231.

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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).

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A245067 Number of three-dimensional random walks with 2n steps in the wedge region x >= y >= z, beginning and ending at the origin without crossing the wedge boundary.

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A241624 Decimal expansion of 'c', one of Polya's Random Walk constants, related to the asymptotics of the number of 3-D random walks starting from and returning to the origin.

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

Formula