Up And Up, Grid Solitaire

This game is played solitaire. (If more than one player wants to play,
then each player plays this game with the same sized grid, and players
compare final scores.)
A player starts with an n-by-n grid drawn on paper. (I suggest an n of
about 8 for beginners.)
The player starts the game by placing a "1" in any square of the grid.
The player, on move number m (m = positive integer), places the number
m in an EMPTY square. Square number m must be either left of, right
of, above, or below square number (m-1), for all m >= 2.
On move 2, the player places the 2 one square from the 1. On move 3,
the player places the 3 two squares from the 2. On move 4, the player
places the 4 three squares from the 3, etc.
Now, k is a variable that increases by 1 on each move, on occasion
being set back to 1 (see below).
A player MUST place the number m a total of k moves from the (m-1), if
the (m-1) was (k-1) moves from the (m-2), UNLESS the player cannot do
so (either because there are no empty squares k squares from the
(m-1), or k squares would be off the grid in all directions).
If a player cannot put an m, for whatever reason, exactly k squares
from square number (m-1), then the count starts over at k=1, and the
player places a number m in an empty square ONE square above, right
of, left of, or below square (m-1).
Then k then becomes 2. And on the next move, the player fills in the
square two squares from the last square, etc.
Play continues until the player cannot move any more, even if k is set
back to 1.
The player's score is the number of squares filled = the last number
written in a square.
Sample game: 6-by-6 grid.
23 * * * * 6
9 15 14 8 10 7
* 3 12 2 11 *
18 16 * 1 17 19
22 * * 21 * 20
24 4 13 * * 5
Score = 24.
Math question: What is the highest possible score for a given n-by-n
grid? Is there an interesting sequence as n = 1,2,3,4, etc... Or is
there an easy pattern to the highest possible scores?
Thanks,
Leroy Quet