[seqfan] « King-walking » integers in a square box

[Eric Angelini]

Hello SeqFans,
In this 4x5 box one can read all
consecutive integers from 0 to 158 
(included):
  5 8 0 7 3 
  9 6 5 1 8 
  1 3 2 4 2 
  4 0 9 7 6
(grid submitted by James Dow Allen
on rec.puzzles two days ago)

The rules are: 
- an integer is there if it's digits
  can be walked on by a chess King 
  (one step in 8 directions: 
   4 straightly, 4 diagonally) 
- two identical digits (or more) can
  follow each other (as if the King 
  was jumping on the same square). 

Example: 
- the integers 58073, 13997 and 13887
  are visible below,
- the integer 159 is not:
  5 8 0 7 3 
  9 6 5 1 8 
  1 3 2 4 2 
  4 0 9 7 6
It seems impossible to find such a 
4x5 box showing all consecutive in-
tegers from 0 to 'n' with 'n' > 158.

Here is Giovanni Resta's 158 solution
for the same box:
  0 3 6 4 2 
  1 7 5 1 3 
  4 0 2 8 9 
  8 9 6 5 7 
(published on rec.puzzles yesterday)  

Question:
Using the same rules, what would be 
the highest reachable integer in the
successive square boxes 1x1, 2x2, 3x3,
4x4, 5x5, ...
This might constitute a seq S for Neil.
[S starts 0, 3, 8, ...]
Best,
É.