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 1-4 of 4 results.

A077818 a(n) is the numerator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

Original entry on oeis.org

40, 190, 15925, 48795, 86221819, 28522360751, 583791967829, 1801511107253, 32337280749408865
Offset: 11

Views

Author

Hugo Pfoertner, Nov 17 2002

Keywords

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.
A more accurate value for this length, determined from a simulation with 27*10^9 walks, is 3953.8+-0.1 (A378903). - Hugo Pfoertner, Dec 15 2024

Examples

			a(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)

References

  • See under A001412.
  • More references are given in the sci.math NG posting in the second link.

Links

Crossrefs

Cf. A001412, A077817, A077819, A077820.
Cf. A377161, A377162, A378903.

Programs

  • Fortran
    c Program provided at first link

Formula

P(n) = a(n) / (5^(n-1) * 3^A077819(n) * 2^A077820(n)) = A377161(n)/A377162(n).

A077819 a(n) is the exponent of 3 in the denominator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 2, 1, 5
Offset: 11

Views

Author

Hugo Pfoertner, Nov 17 2002

Keywords

Examples

			See under A077818.

References

Crossrefs

Cf. A077818, A077820, A377161, A377162, A378903.

Formula

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

A077820 a(n) is the exponent of 2 in the denominator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

Original entry on oeis.org

0, 0, 1, 2, 8, 11, 14, 15, 20
Offset: 11

Views

Author

Hugo Pfoertner, Nov 17 2002

Keywords

Examples

			See under A077818.

References

Crossrefs

Cf. A077818, A077819, A377161, A377162, A378903.

Formula

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

A381979 Decimal expansion of the expected number of steps to termination by self-trapping of a self-avoiding random walk on the square lattice.

Original entry on oeis.org

7, 0, 7, 5, 9
Offset: 2

Views

Author

Yi Yang, Mar 11 2025

Keywords

Comments

The average walk length determined by 1.2*10^12 simulations is 70.75915 +- 0.00010

Examples

			70.759...

References

Links

Crossrefs

Cf. A378903 (The expected walk length on the cubic lattice).
Cf. A077483 (Probability of the occurrence of each walk length).
Cf. A322831.
Showing 1-4 of 4 results.

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