A001412 -id:A001412 - cp's OEIS frontend

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

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

A227338 Number of n-step self-avoiding walks on cubic lattice ending at point with x = k.

Original entry on oeis.org

1, 4, 1, 12, 8, 1, 44, 40, 12, 1, 172, 176, 84, 16, 1, 772, 748, 468, 144, 20, 1, 3308, 3248, 2332, 984, 220, 24, 1, 14924, 14280, 11068, 5756, 1788, 312, 28, 1, 64956, 63768, 51472, 30760, 12108, 2944, 420, 32, 1, 294252, 285296, 237832, 155912, 72948, 22732, 4516

Offset: 0

Joseph Myers, Jul 07 2013

The number of walks ending with x = -k is the same as the number ending with x = k.

			Initial rows (paths of length 0, 1, 2, ...):
1;
4, 1;
12, 8, 1;
44, 40, 12, 1;
...

Bert Dobbelaere, Table of n, a(n) for n = 0..275 (terms 0..152 from Joseph Myers)
J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.

Cf. A000759, A000760, A000761, A000762, A001412.

For n > 0, A001412(n) = T(n,0) + 2 * Sum_{k=1..n} T(n,k). - Bert Dobbelaere, Jan 06 2019

A337400 Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section 2w X 2w where the walk starts at the middle of the tube.

Original entry on oeis.org

6, 26, 6, 98, 30, 6, 330, 146, 30, 6, 1130, 658, 150, 30, 6, 3746, 2858, 722, 150, 30, 6, 12802, 11802, 3450, 726, 150, 30, 6, 42498, 48282, 15930, 3530, 726, 150, 30, 6, 143610, 193714, 72522, 16826, 3534, 726, 150, 30, 6, 472242, 781114, 321794, 80010, 16922, 3534, 726, 150, 30, 6

Offset: 1

Scott R. Shannon, Aug 26 2020

			T(1,2) = 26 as after a step in one of the four directions toward the tube's side the walk must turn along the side; this eliminates the 2-step straight walk in those four directions, so the total number of walks is 6*5 - 4 = 26.
The table begins:
6 26  98 330 1130  3746 12802  42498  143610  472242  1570714   5110426  16779354...
6 30 146 658 2858 11802 48282 193714  781114 3114890 12508114  49767002 199252346...
6 30 150 722 3450 15930 72522 321794 1415450 6134650 26527690 113725546 487875250...
6 30 150 726 3530 16826 80010 373962 1736538 7946946 36158802 162796866 730521658...
6 30 150 726 3534 16922 81274 386138 1833018 8615906 40370370 187477426 867587114...
6 30 150 726 3534 16926 81386 387834 1851546 8780162 41630146 196172338 923017178...
6 30 150 726 3534 16926 81390 387962 1853738 8806962 41893346 198386594 939630954...
6 30 150 726 3534 16926 81390 387966 1853882 8809714 41930594 198788354 943314378...
6 30 150 726 3534 16926 81390 387966 1853886 8809874 41933970 198838482 943903786...
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934146 198842546 943969482...
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842738 943974298...
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974506...
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974510...

Cf. A337401 (start at center of tube's side), A337403 (start at tube's edge), A001412 (w->infinity), A116904, A337023, A259808, A039648.

For n <= w, T(w,n) = A001412(n).

A378903 Decimal expansion of the expected number of steps to termination by self-trapping of a self-avoiding random walk on the cubic lattice.

Original entry on oeis.org

3, 9, 5, 3, 7

Offset: 4

Hugo Pfoertner, Dec 14 2024

See A077818 for more information and links. Since a more accurate value is probably 3953.78..., one should currently use 3953.8 +- 0.1 as a safe estimate.

			3953.7...

Martin Z. Bazant, Topics in Random Walks and Diffusion, Graduate course 18.325, Spring 2001 at the Massachusetts Institute for Technology.
Martin Z. Bazant, Topics in Random Walks and Diffusion, Problem Sets for Spring 2001. In the no longer available solutions to Problem Set 2b, Dion Harmon gave 3960 using 10^5 walks, and E.C.Silva gave 3676 using 1.5*10^4 walks.
Martin Z. Bazant, Problem Set 2 for Graduate course 18.325, local pdf version of postscript file. Problem 5. Self-Trapping Walk.
Hugo Pfoertner, Probability density for the number of steps before trapping occurs, based on 27*10^9 simulated walks (2024).

Cf. A001412, A077817, A077818, A077819, A077820, A377161, A377162.

A000759 Number of n-step self-avoiding walks on cubic lattice ending at point with x=0.

Original entry on oeis.org

1, 4, 12, 44, 172, 772, 3308, 14924, 64956, 294252, 1301044, 5930588, 26506948, 121290940, 546050988, 2505533940, 11340303508, 52147596788, 236995050900, 1091701675948, 4977541017540, 22961416861940, 104965762062612

Offset: 0

N. J. A. Sloane

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).

J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.

Cf. A000760, A000761, A000762, A001412, A227338.

Edited and extended by Joseph Myers, Jul 07 2013

a(17)-a(22) from Bert Dobbelaere, Jan 06 2019

A000760 Number of n-step self-avoiding walks on cubic lattice ending at point with x=1.

Original entry on oeis.org

1, 8, 40, 176, 748, 3248, 14280, 63768, 285296, 1285688, 5794436, 26261224, 119028156, 541876608, 2466620624, 11267536496, 51458718144, 235690960392, 1079212461992, 4953659984000, 22730713367468, 104520944666808

Offset: 1

N. J. A. Sloane

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).

J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.

Cf. A000759, A000761, A000762, A001412, A227338.

Edited and extended by Joseph Myers, Jul 07 2013

a(17)-a(22) from Bert Dobbelaere, Jan 06 2019

A000761 Number of n-step self-avoiding walks on cubic lattice ending at point with x=2.

Original entry on oeis.org

1, 12, 84, 468, 2332, 11068, 51472, 237832, 1095384, 5040568, 23168528, 106496816, 489379904, 2250000884, 10345888480, 47604198576, 219096141188, 1009071461380, 4648802248764, 21431064157200, 98828123716260

Offset: 2

N. J. A. Sloane

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).

J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.

Cf. A000759, A000760, A000762, A001412, A227338.

Edited and extended by Joseph Myers, Jul 07 2013

a(17)-a(22) from Bert Dobbelaere, Jan 06 2019

A000762 Number of n-step self-avoiding walks on cubic lattice ending at point with x=3.

Original entry on oeis.org

1, 16, 144, 984, 5756, 30760, 155912, 766424, 3698848, 17648312, 83558828, 393534176, 1846227984, 8637479208, 40325165648, 187980582568, 875268197452, 4072021100336, 18931821861960, 87979249474568

Offset: 3

N. J. A. Sloane

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).

J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.

Cf. A000759, A000760, A000761, A001412, A227338.

Edited and extended by Joseph Myers, Jul 07 2013

a(17)-a(22) from Bert Dobbelaere, Jan 06 2019

A079156 Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on cubic lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A078717.

Original entry on oeis.org

10, 67, 396, 2201, 11870, 62571, 324896, 1665349, 8457890, 42605267, 213305636, 1061939193, 5263752278, 25984214383, 127848694424, 627084275649, 3067923454498

Offset: 2

Hugo Pfoertner, Dec 29 2002

A conjectured asymptotic behavior for the mean Manhattan displacement is shown in a diagram lim n-> infinity a(n)/(A078717(n)*n^nu)=c, for some values of nu near 0.59 at Pfoertner link

			a(2)=10 because the A078717(2)=5 different self-avoiding 2-step walks end at (1,0,-1),(1,0,1),(1,-1,0),(1,1,0),(2,0,0)->d=2. a(2)=5*2=10. See also "Distribution of end point distance" at Pfoertner link

See under A001412

Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant

Cf. A001412, A078717, A078605 (corresponding square displacement).

c Program for distance counting available at Pfoertner link.

a(n)= sum l=1, A078717(n) (|i_l| + |j_l| + |k_l|) where (i_l, j_l, k_l) are the end points of all different self-avoiding n-step walks starting at (0, 0, 0)

A335806 The number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2x2x2 where the walk starts at the middle of the box's edge.

Original entry on oeis.org

1, 4, 12, 40, 118, 358, 936, 2600, 6212, 16068, 34936, 83708, 163452, 357056, 613592, 1205716, 1770616, 3073480, 3715920, 5573480, 5255048, 6591160, 4353912, 4330096, 1513712, 1061392, 0

Offset: 0

Scott R. Shannon, Aug 14 2020

			a(1) = 4 as the walk is free to move one step in four directions.
a(2) = 12. A first step along either edge leading to the corner leaves two possible second steps. A first step to the centre of either face can be followed by a second step to three edges or to the center of the cube, four steps in all. Thus the total number of 2-step walks is 2*2+2*4 = 12.
a(26) = 0 as it is not possible to visit all 26 available lattice points when the walk starts from the middle of the box's edge.

Cf. A336862 (other box sizes), A337021 (start at center of box), A337033 (start at center of face), A337034 (start at corner of box), A001412, A259808, A039648.

For n>=27 all terms are 0 as the walk contains more steps than there are available lattice points in the 2x2x2 box.

cp's OEIS Frontend

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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A227338 Number of n-step self-avoiding walks on cubic lattice ending at point with x = k.

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A337400 Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section 2w X 2w where the walk starts at the middle of the tube.

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A378903 Decimal expansion of the expected number of steps to termination by self-trapping of a self-avoiding random walk on the cubic lattice.

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A000759 Number of n-step self-avoiding walks on cubic lattice ending at point with x=0.

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A000760 Number of n-step self-avoiding walks on cubic lattice ending at point with x=1.

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A000761 Number of n-step self-avoiding walks on cubic lattice ending at point with x=2.

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A000762 Number of n-step self-avoiding walks on cubic lattice ending at point with x=3.

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A079156 Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on cubic lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A078717.

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A335806 The number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2x2x2 where the walk starts at the middle of the box's edge.

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