A001411 -id:A001411 - cp's OEIS frontend

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

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

Offset: 7

Hugo Pfoertner, Nov 08 2002

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

			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)

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

Hugo Pfoertner, Results for the 2D 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 4, 2002

Cf. A077484, A077482, A001411.

Fortran
```
c See Hugo Pfoertner link.
```

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

A078717 Number of n-step self-avoiding walks on cubic lattice with first step specified.

Original entry on oeis.org

1, 5, 25, 121, 589, 2821, 13565, 64661, 308981, 1468313, 6989025, 33140457, 157329085, 744818613, 3529191009, 16686979329, 78955042017, 372953947349, 1762672203269, 8319554639789, 39285015083693, 185296997240401, 874331369198569

Offset: 1

Hugo Pfoertner, Dec 18 2002

nonn

See A001412 for references.

Hugo Pfoertner, Table of n, a(n) for n = 1..36

Equals A001412/6. Cf. A001411, A046661, A002902.

A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi(2k/n - 1), k = 1..n-1.

Original entry on oeis.org

71, 71, 40, 77, 45, 51, 42, 56, 49, 51, 48, 54

Offset: 3

Hugo Pfoertner, Dec 27 2018

The cases n = 3, 4, and 6 correspond to the usual self-avoiding random walks on the honeycomb net, the square lattice, and the hexagonal lattice, respectively. The other cases n = 5, 7, ... are a generalization using self-avoiding rooted walks similar to those defined in A306175, A306177, ..., A306182. The walk is trapped if it cannot be continued without either hitting an already visited (lattice) point or crossing or touching any straight line connecting successively visited points on the path up to the current point.
The result 71 for n=4 was established in 1984 by Hemmer & Hemmer.
The sequence data are based on the following results of at least 10^9 simulated random walks for each n <= 12, with an uncertainty of +- 0.004 for the average walk length:
   n  length
71.132
70.760 (+-0.001)
40.375
77.150
45.297
51.150
42.049
56.189
48.523
51.486
47.9   (+-0.2)
53.9   (+-0.2)

S. Hemmer, P. C. Hemmer, An average self-avoiding random walk on the square lattice lasts 71 steps, J. Chem. Phys. 81, 584 (1984)
Hugo Pfoertner, Examples of self-trapping random walks.
Hugo Pfoertner, Probability density for the number of steps before trapping occurs, 2018.
Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk.
Alexander Renner, Self avoiding walks and lattice polymers, Diplomarbeit, Universität Wien, December 1994.

Cf. A001668, A001411, A001334, A077482, A306175, A306177, A306178, A306179, A306180, A306181, A306182.

Cf. A122223, A122224, A122226, A127399, A127400, A127401, A300665, A323141, A323560, A323562, A323699.

A342883 Number of n-step self-avoiding walks on 7-D cubic lattice.

Original entry on oeis.org

1, 14, 182, 2366, 30590, 395654, 5110070, 66009062, 852194966, 11002765718, 142019952830, 1833202179662, 23659632189662, 305360673698150, 3940760013826454, 50857078231126286, 656293571739976142, 8469305943784113806, 109290078485661202262, 1410313416278288850230

Offset: 0

N. J. A. Sloane, Mar 31 2021

nonn

N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 10973-11017. Gives terms through a(24).
Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion: tables [Tables in humanly readable form]; Local copy.
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]

For self-avoiding walks on the k-D cubic lattice for k = 2, ..., 12 see A001411, A001412, A010575, A010576, A010577, A342883, A342884, A342885, A342886, A342887, A342888.

A342884 Number of n-step self-avoiding walks on 8-D cubic lattice.

Original entry on oeis.org

1, 16, 240, 3600, 53776, 803504, 11994096, 179054640, 2672126256, 39878886896, 595065468048, 8879592484240, 132491660323472, 1976912303612080, 29496313445323888, 440098575225868624, 6566302628140689744, 97969968518462054352, 1461698348385616122224

Offset: 0

N. J. A. Sloane, Mar 31 2021

nonn

N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 10973-11017. Gives terms through a(24).
Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion: tables [Tables in humanly readable form]; Local copy.
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]

For self-avoiding walks on the k-D cubic lattice for k = 2, ..., 12 see A001411, A001412, A010575, A010576, A010577, A342883, A342884, A342885, A342886, A342887, A342888.

A342885 Number of n-step self-avoiding walks on 9-D cubic lattice.

Original entry on oeis.org

1, 18, 306, 5202, 88146, 1493874, 25300530, 428518386, 7256300850, 122876680626, 2080586127186, 35229409431570, 596495353475538, 10099744526658546, 171003188767881906, 2895335387107970706, 49021668492861718674, 829999403731225961874

Offset: 0

N. J. A. Sloane, Mar 31 2021

nonn

N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 10973-11017. Gives terms through a(24).
Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion: tables [Tables in humanly readable form]; Local copy.
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]

For self-avoiding walks on the k-D cubic lattice for k = 2, ..., 12 see A001411, A001412, A010575, A010576, A010577, A342883, A342884, A342885, A342886, A342887, A342888.

A342886 Number of n-step self-avoiding walks on 10-D cubic lattice.

Original entry on oeis.org

1, 20, 380, 7220, 136820, 2593100, 49121660, 930556460, 17625825740, 333857601020, 6323384122580, 119767717450100, 2268399952520660, 42963566150826380, 813721674662589980, 15411746407417290020, 291893918240586194660, 5528387235193561980740

Offset: 0

N. J. A. Sloane, Mar 31 2021

nonn

N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 10973-11017. Gives terms through a(24).
Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion: tables [Tables in humanly readable form]; Local copy.
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]

For self-avoiding walks on the k-D cubic lattice for k = 2, ..., 12 see A001411, A001412, A010575, A010576, A010577, A342883, A342884, A342885, A342886, A342887, A342888.

A342887 Number of n-step self-avoiding walks on 11-D cubic lattice.

Original entry on oeis.org

1, 22, 462, 9702, 203302, 4260542, 89253582, 1869809502, 39167457582, 820458452462, 17185914925542, 359989506212182, 7540511273930822, 157947298263243742, 3308420553034902382, 69299392385043268822, 1451565583054963249302, 30404929596858248780502

Offset: 0

N. J. A. Sloane, Mar 31 2021

nonn

N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 10973-11017. Gives terms through a(24).
Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion: tables [Tables in humanly readable form]; Local copy.
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]

For self-avoiding walks on the k-D cubic lattice for k = 2, ..., 12 see A001411, A001412, A010575, A010576, A010577, A342883, A342884, A342885, A342886, A342887, A342888.

A342888 Number of n-step self-avoiding walks on 12-D cubic lattice.

Original entry on oeis.org

1, 24, 552, 12696, 291480, 6692424, 153614760, 3526063752, 80931227016, 1857565708968, 42634594787160, 978544945823832, 22459264078075992, 515478463349872200, 11831064537706447464, 271542137952854806776, 6232321082672399260152, 143041632747658763159736

Offset: 0

N. J. A. Sloane, Mar 31 2021

nonn

N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 10973-11017. Gives terms through a(24).
Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion: tables [Tables in humanly readable form]; Local copy.
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]

For self-avoiding walks on the k-D cubic lattice for k = 2, ..., 12 see A001411, A001412, A010575, A010576, A010577, A342883, A342884, A342885, A342886, A342887, A342888.

A002902 Number of n-step self-avoiding walks on a cubic lattice with a first step along the positive x, y, or z axis.

Original entry on oeis.org

3, 15, 75, 363, 1767, 8463, 40695, 193983, 926943, 4404939, 20967075, 99421371, 471987255, 2234455839, 10587573027, 50060937987, 236865126051, 1118861842047, 5288016609807, 24958663919367, 117855045251079, 555890991721203, 2622994107595707

Offset: 1

N. J. A. Sloane

B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 462.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. S. McKenzie and C. Domb, The second osmotic virial coefficient of athermal polymer solutions, Proceedings of the Physical Society, 92 (1967) 632-649.
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
M. F. Sykes, Self-avoiding walks on the simple cubic lattice, J. Chem. Phys., 39 (1963), 410-411.
M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.

Equals (1/2)*A001412. Cf. A078717, A001411, A001413.

Name amended by Scott R. Shannon, Sep 17 2020

cp's OEIS Frontend

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

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A078717 Number of n-step self-avoiding walks on cubic lattice with first step specified.

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A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi*(2*k/n - 1), k = 1..n-1.

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A342883 Number of n-step self-avoiding walks on 7-D cubic lattice.

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A342884 Number of n-step self-avoiding walks on 8-D cubic lattice.

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A342885 Number of n-step self-avoiding walks on 9-D cubic lattice.

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A342886 Number of n-step self-avoiding walks on 10-D cubic lattice.

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A342887 Number of n-step self-avoiding walks on 11-D cubic lattice.

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A342888 Number of n-step self-avoiding walks on 12-D cubic lattice.

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A002902 Number of n-step self-avoiding walks on a cubic lattice with a first step along the positive x, y, or z axis.

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A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi(2k/n - 1), k = 1..n-1.