cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A077483 Numerator of the probability P(n) of the occurrence of a 2D self-trapping walk of length n.

Original entry on oeis.org

2, 5, 31, 173, 1521, 1056, 16709, 184183, 1370009, 474809, 13478513, 150399317, 1034714947, 2897704261
Offset: 7

Views

Author

Hugo Pfoertner, Nov 08 2002

Keywords

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

Examples

			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)

References

  • 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

Crossrefs

Cf. A077484, A077482, A001411.

Programs

  • Fortran
    c See Hugo Pfoertner link.

Formula

P(n) = A077483(n) / ( 3^(n-1) * 2^A077484(n) )

A076874 a(n) = n - floor ( ( 4*n + 1 )^(1/2) ).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59
Offset: 3

Views

Author

Hugo Pfoertner, Nov 24 2002

Keywords

Comments

Conjecture: For n>=7, a(n)-2 is the maximum number of steps in a 2D self-avoiding random walk trapped after n steps having only 2 choices for the next step. a(n) >= A077484(n) + 2.

Links

Crossrefs

Cf. A077483, A077484, A001411.

Programs

A078526 Probability P(n) of the occurrence of a 2D self-trapping walk of length n.

Original entry on oeis.org

1, 5, 31, 173, 1521, 4224, 33418, 184183, 1370009, 3798472, 26957026, 150399317, 1034714947, 2897704261, 19494273755, 109619578524, 724456628891
Offset: 7

Views

Author

Hugo Pfoertner, Nov 27 2002

Keywords

Comments

This is a cleaner representation than the one given by A077483 and A077484, using the upper bound for the denominator A077484 given in A076874.

Examples

			See under A077483; the inclusion of a(7)=1 is somewhat artificial due to the occurrence of 2^(-1) in the denominator: P(7)=a(7)/(3^6 *2^(7-floor(sqrt(29))-3))= 1/(729*2^(7-5-3))=1/(729*2*(-1))=2/729 See also: "Count self-trapping walks up to length 23" provided at given link.

References

Links

Crossrefs

Cf. A077483, A077484, A076874, A001411.

Programs

  • Fortran
    c Program provided at given link

Formula

P(n) = a(n) / ( 3^(n-1) * 2^(n-floor((4*n+1)^(1/2))-3) ) = a(n) / ( 3^(n-1) * 2^(A076874(n)-3) )
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.