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#10 by Alois P. Heinz at Mon May 27 11:38:48 EDT 2024
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#9 by Robert C. Lyons at Mon May 27 11:20:52 EDT 2024
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#8 by Robert C. Lyons at Mon May 27 11:20:50 EDT 2024
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| PROG
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FORTRAN(Fortran) c programProgram provided at first link
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| STATUS
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approved
editing
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#7 by Russ Cox at Sat Mar 31 10:29:01 EDT 2012
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| AUTHOR
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_Hugo Pfoertner (hugo(AT)pfoertner.org), _, Nov 08 2002
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Discussion
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| Sat Mar 31
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| OEIS Server: https://oeis.org/edit/global/581
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#6 by T. D. Noe at Wed Aug 10 20:27:59 EDT 2011
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#5 by Hugo Pfoertner at Wed Aug 10 16:26:56 EDT 2011
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#4 by Hugo Pfoertner at Wed Aug 10 16:26:48 EDT 2011
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| LINKS
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Hugo Pfoertner, <a href="httphttps://mathforumgroups.google.orgcom/discussgroup/sci.math/tmsg/394788205ce31e504d1a0a">Self-trapping random walks on square lattice in 2-D (cubic in 3-D).Posting in NG sci.math dated March 4, 2002</a>
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| STATUS
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approved
editing
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#3 by N. J. A. Sloane at Tue Jul 19 03:00:00 EDT 2005
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| PROG
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FortranFORTRAN program provided at first link
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| KEYWORD
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frac,more,nonn,walk,new
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#2 by N. J. A. Sloane at Thu Feb 19 03:00:00 EST 2004
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| KEYWORD
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frac,more,nonn,walk,new
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| AUTHOR
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Hugo Pfoertner (allhugo(AT)abouthugopfoertner.deorg), Nov 08 2002
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#1 by N. J. A. Sloane at Fri May 16 03:00:00 EDT 2003
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| NAME
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Probability P(n) of the occurrence of a 2D self-trapping walk of length n: Numerator.
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| DATA
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2, 5, 31, 173, 1521, 1056, 16709, 184183, 1370009, 474809, 13478513, 150399317, 1034714947, 2897704261
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| OFFSET
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7,1
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| COMMENTS
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A comparison of the exact probabilities with simulation results obtained for 1.2*10^10 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 for n=31 (P(31)~=1/85.01) and drops exponentially for large n (P(800)~=1/10^9). The average walk length determined by the numerical simulation is sum n=7..infinity (n*P(n))=70.7598+-0.001
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| REFERENCES
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See under A001411
Alexander Renner: Self avoiding walks and lattice polymers. Diplomarbeit University of Vienna, December 1994
More references are given in the sci.math NG posting in the second link
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| LINKS
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Hugo Pfoertner, <a href="http://www.randomwalk.de/stw2d.html">Results for the 2D Self-Trapping Random Walk</a>
Hugo Pfoertner, <a href="http://mathforum.org/discuss/sci.math/t/394788">Self-trapping random walks on square lattice in 2-D (cubic in 3-D).Posting in NG sci.math dated March 4, 2002</a>
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| FORMULA
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P(n) = a077483(n) / ( 3^(n-1) * 2^a077484(n) )
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| EXAMPLE
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A077483(10)=173 and A077484(10)=1 because there are 4 different probabilities for the 50 (=2*A077482(10)) walks: 4 walks with probability p1=1/6561, 14 walks with p2=1/8748, 22 walks with p3=1/13122, 10 walks with p4=1/19683. The sum of all probabilities is P(10) = 4*p1+14*p2+22*p3+10*p4 = (12*4+9*14+6*22+4*10)/78732 = 346/78732 = 173 / (3^9 * 2^1)
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| PROG
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Fortran program provided at first link
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| CROSSREFS
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Cf. A077484, A077482, A001411.
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| KEYWORD
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frac,more,nonn,walk
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| AUTHOR
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Hugo Pfoertner (all(AT)abouthugo.de), Nov 08 2002
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| STATUS
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approved
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