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

1,2

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.

Links

Table of n, a(n) for n=1..9.

Giovanni Resta, Illustration of a(2)-a(9)

Gordon Hamilton, $1,000,000 Unsolved Problem for Grade 8 (2011)

Formula

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

Example

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  .  .  .  .  .  .
-------------------------

Crossrefs

Cf. A226595.

Sequence in context: A033627 A066512 A304116 * A135678 A001195 A295513

Adjacent sequences: A226593 A226594 A226595 * A226597 A226598 A226599

Keyword

nonn,hard,more

Author

Charles R Greathouse IV and Giovanni Resta, Jun 13 2013

Status

approved