Search: id:A077483
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%I A077483
%S A077483 2,5,31,173,1521,1056,16709,184183,1370009,474809,13478513,150399317,
%T A077483 1034714947,2897704261
%N A077483 Probability P(n) of the occurrence of a 2D self-trapping walk of length 
               n: Numerator.
%C A077483 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
%D A077483 See under A001411
%D A077483 Alexander Renner: Self avoiding walks and lattice polymers. Diplomarbeit 
               University of Vienna, December 1994
%D A077483 More references are given in the sci.math NG posting in the second link
%H A077483 Hugo Pfoertner, <a href="http://www.randomwalk.de/stw2d.html">Results 
               for the 2D Self-Trapping Random Walk</a>
%H A077483 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>
%F A077483 P(n) = a077483(n) / ( 3^(n-1) * 2^a077484(n) )
%e A077483 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)
%o A077483 FORTRAN program provided at first link
%Y A077483 Cf. A077484, A077482, A001411.
%Y A077483 Sequence in context: A000133 A059086 A107389 this_sequence A119242 A068145 
               A032112
%Y A077483 Adjacent sequences: A077480 A077481 A077482 this_sequence A077484 A077485 
               A077486
%K A077483 frac,more,nonn,walk
%O A077483 7,1
%A A077483 Hugo Pfoertner (hugo(AT)pfoertner.org), Nov 08 2002

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