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A077818 Probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n: numerator. 3
40, 190, 15925, 48795, 86221819, 28522360751, 583791967829, 1801511107253, 32337280749408865 (list; graph; refs; listen; history; text; internal format)
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.

REFERENCES

See under A001412.

More references are given in the sci.math NG posting in the second link.

LINKS

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

FORMULA

P(n) = A077818(n) / ( 5^(n-1) * 3^A077819(n) * 2^A077820(n) )

EXAMPLE

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)

PROG

FORTRAN program provided at first link

CROSSREFS

Cf. A001412, A077817, A077819, A077820.

Sequence in context: A251203 A187510 A187379 * A247405 A235270 A181637

Adjacent sequences: A077815 A077816 A077817 * A077819 A077820 A077821

KEYWORD

frac,nonn

AUTHOR

Hugo Pfoertner, Nov 17 2002

STATUS

approved

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Last modified January 28 14:40 EST 2023. Contains 359895 sequences. (Running on oeis4.)