A226595 Lengths of maximal nontouching increasing paths in n X n grids.

0, 2, 4, 7, 9, 12, 15, 17, 20, 24, 27, 29, 33, 36, 39

Offset: 1

Author

Charles R Greathouse IV and Giovanni Resta, Jun 13 2013

Keywords

• nonn
• hard
• nice
• more

Comments

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

Examples

			An example for a(8)=17
-------------------------
01 02  .  .  .  . 05  .
..  . 03  . 04  .  . 18
09 07  .  . 06  .  . 16
..  .  .  .  .  .  . 14
..  .  .  .  .  .  . 12
..  . 08  .  .  .  .  .
10 17 15 13  .  .  .  .
..  .  .  . 11  .  .  .
-------------------------
		

Links

• Giovanni Resta, Illustration of a(2)-a(9)
• Bert Dobbelaere, C++ program
• Gordon Hamilton, $1,000,000 Unsolved Problem for Grade 8 (2011)
• Stan Wagon, Illustration for a(2)-a(12)

Crossrefs

Cf. A226596.

Formula

a(n) <= A226596(n) <= A160663(n-1).

Extensions

a(10)-a(12) from Joseph DeVincentis via Charles R Greathouse IV, Oct 08 2015

a(13) from Charles R Greathouse IV, Oct 25 2015 using code from Joseph DeVincentis

a(14)-a(15) from Bert Dobbelaere, Oct 05 2018