A306179 -id:A306179 - cp's OEIS frontend

A306178 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/4, +-Pi/2, +-3*Pi/4.

1, 4, 15, 86, 492, 2992, 18172, 110643, 672267, 4069122, 24578785, 147972210, 889332713, 5331980703, 31924424199

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular octagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306179, A306180, A306181, A306182.

a(7)-a(15) from Bert Dobbelaere, May 15 2025

A306175 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/5, +-3*Pi/5.

Original entry on oeis.org

1, 2, 6, 19, 66, 229, 831, 2991, 10859, 39173, 141631, 510079, 1835583, 6586932, 23614821, 84492315, 302014619, 1077860479, 3843695976, 13690026688

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular pentagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 5.

Cf. A266925, A037245, A151541, A306177, A306178, A306179, A306180, A306181, A306182, A316195.

a(7)-a(9) from Hugo Pfoertner, Dec 23 2018

a(10)-a(20) from Bert Dobbelaere, May 15 2025

A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3Pi/7, +-5Pi/7.

Original entry on oeis.org

1, 3, 11, 55, 275, 1444, 7609, 40220, 212051, 1114206, 5840287, 30521627, 159166728, 828130291, 4301636648

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular heptagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 4.

Cf. A266925, A037245, A306175, A151541, A306178, A306179, A306180, A306181, A306182.

a(7)-a(15) from Bert Dobbelaere, May 15 2025

A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2Pi/5, +-3Pi/5, +-4*Pi/5.

Original entry on oeis.org

1, 5, 23, 169, 1233, 9551

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular decagon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306181, A306182.

A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9Pi/11.

Original entry on oeis.org

1, 5, 28, 235, 1970, 17201, 149420, 1295637, 11178026, 96047288, 822418731, 7020762655

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular 11-gon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306180, A306182.

a(7)-a(12) from Bert Dobbelaere, May 15 2025

A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2Pi/6, +-3Pi/6, +-4Pi/6, +-5Pi/6.

Original entry on oeis.org

1, 6, 33, 306, 2765, 26290, 247737, 2332965, 21856232, 204019196, 1897940592, 17606864337

Offset: 1

Hugo Pfoertner, Jun 23 2018

The turning angles are those of the regular 12-gon, with one vertex corresponding to the forbidden U-turn. The path may neither intersect nor touch itself anywhere.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of walks of lengths <= 3.

Cf. A266925, A037245, A306175, A151541, A306177, A306178, A306179, A306180, A306181.

a(7)-a(12) from Bert Dobbelaere, May 15 2025

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.

A316199 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/9, +-3Pi/9, +-5Pi/9, +-7Pi/9, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 8, 17, 472, 7042

Offset: 1

Hugo Pfoertner, Jul 04 2018

Holes are excluded, i.e., the boundary path may nowhere touch or intersect itself.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.
Hugo Pfoertner, Illustration of polygons of perimeter <= 12.

Cf. A306179, A316195, A316197, A316198.

A346129 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/9, +-Pi/3, +-5Pi/9, +-7Pi/9, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41

Offset: 1

Hugo Pfoertner and Markus Sigg, Aug 01 2021

Although closed walks (see A316199) would be allowed, no closed walk that fits into a smaller enclosing circle than any open walk of the same length is known.

			See link for illustrations of terms corresponding to diameters D < 3.50.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A127399, A127400, A127401, A306179, A316199.

Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

cp's OEIS Frontend

A306178 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/4, +-Pi/2, +-3*Pi/4.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A306175 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/5, +-3*Pi/5.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3*Pi/7, +-5*Pi/7.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2*Pi/5, +-3*Pi/5, +-4*Pi/5.

Views

Author

Keywords

Comments

Links

Crossrefs

A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3*Pi/11, +-5*Pi/11, +-7*Pi/11, +-9*Pi/11.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2*Pi/6, +-3*Pi/6, +-4*Pi/6, +-5*Pi/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

A316199 Number of self-avoiding polygons with perimeter 2*n and sides = 1 that have vertex angles from the set +-Pi/9, +-3*Pi/9, +-5*Pi/9, +-7*Pi/9, not counting rotations and reflections as distinct.

Views

Author

Keywords

Comments

Links

Crossrefs

A346129 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/9, +-Pi/3, +-5*Pi/9, +-7*Pi/9, of length m + 1 fits into the smallest circle that can enclose a walk of length m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A306177 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/7, +-3Pi/7, +-5Pi/7.

A306180 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/5, +-2Pi/5, +-3Pi/5, +-4*Pi/5.

A306181 Number of unrooted self-avoiding walks with n steps that can make turns from the set +-Pi/11, +-3Pi/11, +-5Pi/11, +-7Pi/11, +-9Pi/11.

A306182 Number of unrooted self-avoiding walks with n steps that can make turns from the set 0, +-Pi/6, +-2Pi/6, +-3Pi/6, +-4Pi/6, +-5Pi/6.

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.

A316199 Number of self-avoiding polygons with perimeter 2n and sides = 1 that have vertex angles from the set +-Pi/9, +-3Pi/9, +-5Pi/9, +-7Pi/9, not counting rotations and reflections as distinct.

A346129 Numbers m such that no self-avoiding walk that can make turns from the set +-Pi/9, +-Pi/3, +-5Pi/9, +-7Pi/9, of length m + 1 fits into the smallest circle that can enclose a walk of length m.