A226596 Lengths of maximal non-crossing and non-overlapping increasing paths in n X n grids.

0, 2, 4, 7, 10, 13, 16, 20, 22

Offset: 1

Author

Charles R Greathouse IV and Giovanni Resta, Jun 13 2013

Keywords

• nonn
• hard
• more

Comments

The path is allowed to touch but not cross itself on single points, but not on segments of any length. "Increasing" means that the (Euclidean) length of each line segment must be strictly longer than the last.

Examples

			A solution for the case a(8)=20 is
-------------------------
01 02  .  .  .  .  . 16
..  . 03  .  .  .  . 14
09  . 15  . 05  .  . 12
..  . 04  .  .  .  .  .
..  . 06 13  . 07  . 21
..  . 08  . 11  .  . 19
10  .  .  .  .  .  . 17
20 18  .  .  .  .  .  .
-------------------------
		

Links

• Giovanni Resta, Illustration of a(2)-a(9)
• Gordon Hamilton, $1,000,000 Unsolved Problem for Grade 8 (2011)

Crossrefs

Cf. A226595.

Formula

a(n) <= A160663(n-1).