A377161 -id:A377161 - cp's OEIS frontend

A377162 a(n) is the denominator corresponding to A377161(n).

1953125, 9765625, 58593750, 976562500, 4687500000000, 1687500000000000, 22500000000000000, 75000000000000000, 194400000000000000000

Offset: 11

Hugo Pfoertner, Oct 20 2024

			A377161(n)/a(n) for n = 11, 12, ... : 8/1953125, 38/9765625, 637/58593750, 9759/976562500, 86221819/4687500000000, 28522360751/1687500000000000, 583791967829/22500000000000000, ...

A377161 are the corresponding numerators.

See A077818 for more information.

a(n) = (A377161(n)/A077818(n)) * (5^(n-1) * 3^A077819(n) * 2^A077820(n)).

A077817 Number of self-avoiding walks on the cubic lattice trapped after n steps.

Original entry on oeis.org

5, 20, 229, 921, 7156, 29567, 193932, 821797, 4902336, 21201528, 119162697, 523550761

Offset: 11

Hugo Pfoertner, Nov 17 2002

Only 1/48 of all possible walks is counted by selecting the first step in +x direction and requiring the first steps changing y and z to be positive, with the first +y step before the first +z step.

See references given for A001412

Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk.

Cf. A001412, A077818, A077819, A077820, A377161, A377162.

Fortran
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c Program provided at given link
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a(20)-a(22) from Bert Dobbelaere, Mar 23 2025

A077818 a(n) is the numerator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

Original entry on oeis.org

40, 190, 15925, 48795, 86221819, 28522360751, 583791967829, 1801511107253, 32337280749408865

Offset: 11

Hugo Pfoertner, Nov 17 2002

A comparison of the exact probabilities with simulation results obtained for 1.1*10^9 random walks is given under "Results of simulation, comparison with exact probabilities" in the first link. The behavior of P(n) for larger values of n is illustrated in "Probability density for the number of steps before trapping occurs" at the same location. P(n) has a maximum around n~=600 (P(600)~=1/4760) and drops exponentially for large n (P(45000)~=1/10^9). The average walk length determined by the numerical simulation is sum n=11..infinity (n*P(n))=3953.65 +-0.20.

A more accurate value for this length, determined from a simulation with 27*10^9 walks, is 3953.8+-0.1 (A378903). - Hugo Pfoertner, Dec 15 2024

			a(13)=15925, A077819(13)=A077820(13)=1 because there are 5 different probabilities for the 1832 (=8*A077817(13)) walks: 256 walks with probability p1=1/125000000, 88 with p2=1/146484375, 600 with p3=1/156250000, 728 with p4=1/146484375 and 160 with p5=1/244140625. P(13)=256*p1+88*p2+600*p3+728*p4+160*p5=637/(6*5^10)=25*637/(5^12*6)= 15295/(5^(13-1)*3^1*2^1)

See under A001412.
More references are given in the sci.math NG posting in the second link.

Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk.
Hugo Pfoertner, Self-trapping random walks on square lattice in 2-D (cubic in 3-D). Posting in NG sci.math dated March 5, 2002.
Alexander Renner, Self avoiding walks and lattice polymers, Diplomarbeit, Universität Wien, December 1994.

Cf. A001412, A077817, A077819, A077820.

Cf. A377161, A377162, A378903.

Fortran
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c Program provided at first link
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P(n) = a(n) / (5^(n-1) * 3^A077819(n) * 2^A077820(n)) = A377161(n)/A377162(n).

A077819 a(n) is the exponent of 3 in the denominator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 2, 1, 5

Offset: 11

Hugo Pfoertner, Nov 17 2002

			See under A077818.

See under A077818.

Cf. A077818, A077820, A377161, A377162, A378903.

P(n) = A077818(n) / ( 5^(n-1) * 3^a(n) * 2^A077820(n) ).

A077820 a(n) is the exponent of 2 in the denominator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

Original entry on oeis.org

0, 0, 1, 2, 8, 11, 14, 15, 20

Offset: 11

Hugo Pfoertner, Nov 17 2002

			See under A077818.

See under A077818.

Cf. A077818, A077819, A377161, A377162, A378903.

P(n) = A077818(n) / ( 5^(n-1) * 3^A077819(n) * 2^a(n) ).

A378903 Decimal expansion of the expected number of steps to termination by self-trapping of a self-avoiding random walk on the cubic lattice.

Original entry on oeis.org

3, 9, 5, 3, 7

Offset: 4

Hugo Pfoertner, Dec 14 2024

See A077818 for more information and links. Since a more accurate value is probably 3953.78..., one should currently use 3953.8 +- 0.1 as a safe estimate.

			3953.7...

Martin Z. Bazant, Topics in Random Walks and Diffusion, Graduate course 18.325, Spring 2001 at the Massachusetts Institute for Technology.
Martin Z. Bazant, Topics in Random Walks and Diffusion, Problem Sets for Spring 2001. In the no longer available solutions to Problem Set 2b, Dion Harmon gave 3960 using 10^5 walks, and E.C.Silva gave 3676 using 1.5*10^4 walks.
Martin Z. Bazant, Problem Set 2 for Graduate course 18.325, local pdf version of postscript file. Problem 5. Self-Trapping Walk.
Hugo Pfoertner, Probability density for the number of steps before trapping occurs, based on 27*10^9 simulated walks (2024).

Cf. A001412, A077817, A077818, A077819, A077820, A377161, A377162.

cp's OEIS Frontend

A377162 a(n) is the denominator corresponding to A377161(n).

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A077817 Number of self-avoiding walks on the cubic lattice trapped after n steps.

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A077818 a(n) is the numerator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

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A077819 a(n) is the exponent of 3 in the denominator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

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A077820 a(n) is the exponent of 2 in the denominator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

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A378903 Decimal expansion of the expected number of steps to termination by self-trapping of a self-avoiding random walk on the cubic lattice.

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