login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A077483 Probability P(n) of the occurrence of a 2D self-trapping walk of length n: Numerator. 5
2, 5, 31, 173, 1521, 1056, 16709, 184183, 1370009, 474809, 13478513, 150399317, 1034714947, 2897704261 (list; graph; refs; listen; history; text; internal format)
OFFSET
7,1
COMMENTS
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
REFERENCES
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
LINKS
FORMULA
P(n) = a077483(n) / ( 3^(n-1) * 2^a077484(n) )
EXAMPLE
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)
PROG
FORTRAN program provided at first link
CROSSREFS
Sequence in context: A107389 A261750 A189559 * A119242 A068145 A032112
KEYWORD
frac,more,nonn,walk
AUTHOR
Hugo Pfoertner, Nov 08 2002
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 23 07:16 EDT 2024. Contains 371905 sequences. (Running on oeis4.)