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A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi*(2*k/n - 1), k = 1..n-1.

%I A322831 #24 Apr 27 2025 05:12:08
%S A322831 71,71,40,77,45,51,42,56,49,51,48,54
%N A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi*(2*k/n - 1), k = 1..n-1.
%C A322831 The cases n = 3, 4, and 6 correspond to the usual self-avoiding random walks on the honeycomb net, the square lattice, and the hexagonal lattice, respectively. The other cases n = 5, 7, ... are a generalization using self-avoiding rooted walks similar to those defined in A306175, A306177, ..., A306182. The walk is trapped if it cannot be continued without either hitting an already visited (lattice) point or crossing or touching any straight line connecting successively visited points on the path up to the current point.
%C A322831 The result 71 for n=4 was established in 1984 by Hemmer & Hemmer.
%C A322831 The sequence data are based on the following results of at least 10^9 simulated random walks for each n <= 12, with an uncertainty of +- 0.004 for the average walk length:
%C A322831    n  length
%C A322831    3  71.132
%C A322831    4  70.760 (+-0.001)
%C A322831    5  40.375
%C A322831    6  77.150
%C A322831    7  45.297
%C A322831    8  51.150
%C A322831    9  42.049
%C A322831   10  56.189
%C A322831   11  48.523
%C A322831   12  51.486
%C A322831   13  47.9   (+-0.2)
%C A322831   14  53.9   (+-0.2)
%H A322831 S. Hemmer, P. C. Hemmer, <a href="https://doi.org/10.1063/1.447349">An average self-avoiding random walk on the square lattice lasts 71 steps</a>, J. Chem. Phys. 81, 584 (1984)
%H A322831 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/A322831.htm">Examples of self-trapping random walks</a>.
%H A322831 Hugo Pfoertner, <a href="/A322831/a322831.pdf">Probability density for the number of steps before trapping occurs</a>, 2018.
%H A322831 Hugo Pfoertner, <a href="http://www.randomwalk.de/stw2d.html">Results for the 2D Self-Trapping Random Walk.</a>
%H A322831 Alexander Renner, <a href="http://www.tbi.univie.ac.at/papers/Abstracts/alex_dipl.pdf">Self avoiding walks and lattice polymers</a>, Diplomarbeit, Universität Wien, December 1994.
%Y A322831 Cf. A001668, A001411, A001334, A077482, A306175, A306177, A306178, A306179, A306180, A306181, A306182.
%Y A322831 Cf. A122223, A122224, A122226, A127399, A127400, A127401, A300665, A323141, A323560, A323562, A323699.
%K A322831 nonn,more
%O A322831 3,1
%A A322831 _Hugo Pfoertner_, Dec 27 2018