OFFSET
11,1
COMMENTS
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
REFERENCES
See under A001412.
More references are given in the sci.math NG posting in the second link.
LINKS
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.
EXAMPLE
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)
PROG
(Fortran) c Program provided at first link
CROSSREFS
KEYWORD
nonn,frac,walk,hard,more
AUTHOR
Hugo Pfoertner, Nov 17 2002
STATUS
approved