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The extension of the contest problem to larger sets of hinge angles was proposed by James R. Buddenhagen. A link to the contest rules is given in A127400. Results up to n=32 were found by Markus Sigg. Known lower bounds for the next terms are a(27)>=29, a(28)>=32, a(29)>=34, a(30)>=34, a(31)>=34, a(32)>=39.

a(n)>=max(A053416(n),A053479(n),A053417(n)). a(n) is an upper bound for the number of segments of a self avoiding path on the 2-dimensional triangular lattice such that the path fits into a circle of diameter n. A122226(n)<=a(n).

The problems corresponding to n=3,4,6 had been excluded from the contest.

a(16) >= 52. - Hugo Pfoertner, Jun 30 2021

Links to the contest rules and to visualizations of the results are given in A127400. Known lower bounds for the next terms are a(10)>=50, a(11)>=77, a(12)>=71. [updated by Hugo Pfoertner, May 21 2011]

The path may be open or closed. For larger n several solutions with the same number of segments exist.

It is conjectured that a(n) >= A123690(n)-1, i.e., that it is always possible to find a path visiting all grid points covered by a circle, irrespective of the position of its center. - Hugo Pfoertner, Mar 02 2018

The path may be open or closed. For larger n several solutions with the same number of segments exist. An upper bound for the number of segments is given by A127406(n).

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)

Open and closed walks are allowed. It is conjectured that all optimal paths are closed except for the trivial path of length 1. See the related conjecture in A122226.