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A077818
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Probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n: numerator.
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3
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OFFSET
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11,1
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COMMENTS
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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.
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REFERENCES
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See under A001412.
More references are given in the sci.math NG posting in the second link.
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LINKS
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Table of n, a(n) for n=11..19.
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
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FORMULA
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P(n) = A077818(n) / ( 5^(n-1) * 3^A077819(n) * 2^A077820(n) )
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EXAMPLE
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A077818(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)
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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. A001412, A077817, A077819, A077820.
Sequence in context: A251203 A187510 A187379 * A247405 A235270 A181637
Adjacent sequences: A077815 A077816 A077817 * A077819 A077820 A077821
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KEYWORD
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frac,nonn
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AUTHOR
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Hugo Pfoertner, Nov 17 2002
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STATUS
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approved
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