A001412 -id:A001412 - cp's OEIS frontend

A077817 Number of self-avoiding walks on the cubic lattice trapped after n steps.

5, 20, 229, 921, 7156, 29567, 193932, 821797, 4902336, 21201528, 119162697, 523550761

Offset: 11

Hugo Pfoertner, Nov 17 2002

Only 1/48 of all possible walks is counted by selecting the first step in +x direction and requiring the first steps changing y and z to be positive, with the first +y step before the first +z step.

See references given for A001412

Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk.

Cf. A001412, A077818, A077819, A077820, A377161, A377162.

Fortran
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c Program provided at given link
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a(20)-a(22) from Bert Dobbelaere, Mar 23 2025

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

Hugo Pfoertner, Nov 17 2002

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

			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)

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

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.
Alexander Renner, Self avoiding walks and lattice polymers, Diplomarbeit, Universität Wien, December 1994.

Cf. A001412, A077817, A077819, A077820.

Cf. A377161, A377162, A378903.

Fortran
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c Program provided at first link
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P(n) = a(n) / (5^(n-1) * 3^A077819(n) * 2^A077820(n)) = A377161(n)/A377162(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.

A118313 Sum of squared end-to-end distances of all n-step self-avoiding walks on the simple cubic lattice.

Original entry on oeis.org

0, 6, 72, 582, 4032, 25566, 153528, 886926, 4983456, 27401502, 148157880, 790096950, 4166321184, 21760624254, 112743796632, 580052260230, 2966294589312, 15087996161382, 76384144381272, 385066579325550, 1933885653380544, 9679153967272734, 48295148145655224, 240292643254616694, 1192504522283625600, 5904015201226909614, 29166829902019914840, 143797743705453990030, 707626784073985438752, 3476154136334368955958, 17048697241184582716248, 83487969681726067169454, 408264709609407519880320, 1993794711631386183977574, 9724709261537887936102872, 47376158929939177384568598, 230547785968352575619933376

Offset: 0

R. J. Mathar, May 14 2006

nonn

Number of walks is A001412(n).
a(5) is 25556 according to MacDonald et al., but 25566 according to Clisby et al. and is therefore conjectural for now. - R. J. Mathar, Aug 31 2007
Confirmed that a(5) is 25566 [from Nathan Clisby]. Right-hand column, table, p.5 of Schram.

R. D. Schram, G. T. Barkema, R. H. Bisseling, Table of n, a(n) for n = 0..36
N. Clisby, R. Liang and G. Slade Self-avoiding walk enumeration via the lace expansion J. Phys. A: Math. Theor. vol. 40 (2007) p 10973-11017, Table A5 for n<=30.
A. J. Guttmann, On the critical behavior of self-avoiding walks, J. Phys. A 20 (1987), 1839-1854.
D. MacDonald, S. Joseph, D. L. Hunter, L. L. Mosley, N. Jan and A. J. Guttmann, Self-avoiding walks on the simple cubic lattice,J Phys A: Math Gen 33 (2000) No 34, 5973-5983
Raoul D. Schram, Gerard T. Barkema, Rob H. Bisseling, Exact enumeration of self-avoiding walks, J Stat. Mech. (2011) P06019.

Cf. A001412, A078605, A079156.

a(5) corrected by Nathan Clisby, Nov 24 2010

a(14), a(22) corrected by Hugo Pfoertner, Aug 13 2011

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.

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A077817 Number of self-avoiding walks on the cubic lattice trapped after n steps.

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A077818 a(n) is the numerator of the probability P(n) of the occurrence of a 3-dimensional 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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A118313 Sum of squared end-to-end distances of all n-step self-avoiding walks on the simple cubic lattice.

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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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