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A hexagonal board of edge-length 3, for example, is numbered spirally as:

.

17--18--19

/

16 6---7---8

/ / \

15 5 1---2 9

\ \ / /

14 4---3 10

\ /

13--12--11

.

In Glinski's hexagonal chess, a knight (N) can move to these (o) cells:

.

. . . . .

. . o o . .

. o . . . o .

. o . . . . o .

. . . . N . . . .

. o . . . . o .

. o . . . o .

. . o o . .

. . . . .

.

a(n) stays constant at 72085 for n >= 177 since 72085 is also the last cell visited by knight moves on a spirally numbered infinite hexagonal board, moving to the lowest unvisited cell at each step.

For a knight moving on a spirally numbered hexagonal board to the lowest available unvisited cell, see A327131, a(n) gives the number of steps before the knight is trapped when the knight starts on the cell numbered n.

See A327131 for the allowed knight moves, a diagram of the hexagonal board, and an illustration of the knight's path for n = 1.

For the first 100000 terms the longest path before the knight is trapped is for starting starting cell 81479 where it is trapped after 8125572 steps, the final cell being 8085793. In the same range the shortest path before being trapped is for starting cell 1036 where it is trapped after 1603 steps, the final cell being 1267. See the image in the links. This is likely the shortest path to being trapped for all starting cells.

For a knight moving on a spirally numbered hexagonal board to the lowest available unvisited cell, see A327131, a(n) gives the cell number where the knight is trapped when the knight starts on the cell numbered n.

See A327131 for the allowed knight moves, a diagram of the hexagonal board, and an illustration of the knight's path for n = 1.

For the first 100000 terms the largest cell number where the knight is trapped is for starting starting cell 81479 where the final cell has number 8085793, being reached after 8125572 steps. In the same range the smallest cell number where the night is trapped is for starting cell 1036 where the final cell has number 1267, being reached after 1603 steps. See A333683 for an image of this path.