cp's OEIS Frontend

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.

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.

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

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.

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.

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.

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

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

3 71.132

4 70.760 (+-0.001)

5 40.375

6 77.150

7 45.297

8 51.150

9 42.049

10 56.189

11 48.523

12 51.486

13 47.9 (+-0.2)

14 53.9 (+-0.2)

Paths which are equivalent under rotation, reflection or reversal are counted only once.

A path is considered as symmetric if its "spine", i.e., the connection of the end points of the moves by straight lines, has mirror or point symmetry. The non-symmetric details of a single move are ignored.