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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.

%I A077818 #34 Dec 15 2024 07:24:26
%S A077818 40,190,15925,48795,86221819,28522360751,583791967829,1801511107253,
%T A077818 32337280749408865
%N A077818 a(n) is the numerator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.
%C A077818 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.
%C A077818 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
%D A077818 See under A001412.
%D A077818 More references are given in the sci.math NG posting in the second link.
%H A077818 Hugo Pfoertner, <a href="https://www.randomwalk.de/stw3d.html">Results for the 3-dimensional Self-Trapping Random Walk</a>.
%H A077818 Hugo Pfoertner, <a href="https://groups.google.com/g/sci.math/c/awqJHabD44I/m/ChpNUB7jXCAJ">Self-trapping random walks on square lattice in 2-D (cubic in 3-D)</a>. Posting in NG sci.math dated March 5, 2002.
%H A077818 Alexander Renner, <a href="https://www.tbi.univie.ac.at/papers/Abstracts/alex_dipl.pdf">Self avoiding walks and lattice polymers</a>, Diplomarbeit, Universität Wien, December 1994.
%F A077818 P(n) = a(n) / (5^(n-1) * 3^A077819(n) * 2^A077820(n)) = A377161(n)/A377162(n).
%e A077818 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)
%o A077818 (Fortran) c Program provided at first link
%Y A077818 Cf. A001412, A077817, A077819, A077820.
%Y A077818 Cf. A377161, A377162, A378903.
%K A077818 nonn,frac,walk,hard,more
%O A077818 11,1
%A A077818 _Hugo Pfoertner_, Nov 17 2002