A394363 Squares visited by a knight moving on a square-spiral numbered board, starting from square 1, always moving to the lowest available unvisited square while never crossing the path of any previous step.

1, 10, 3, 6, 9, 22, 19, 4, 13, 28, 25, 46, 21, 40, 5, 14, 31, 54, 27, 48, 51, 80, 47, 76, 43, 70, 39, 16, 33, 58, 55, 86, 125, 50, 85, 26, 53

Offset

1,2

Comments

This is a variation of A316667, but here as the knight moves it leaves a straight line between the center of the two squares it moves from and to, and subsequent steps cannot cross the lines between squares of any previous steps.

The path is finite - after visiting 37 squares, the square with number 53 is reached, after which all 8 squares the knight could move to have either been previously visited or are blocked by the path of previous steps. See the attached image.

Links

Table of n, a(n) for n=1..37.

Scott R. Shannon, Image of the full path. The starting square is green, the last square is red, and the eight squares surrounding the last square that have been previously visited or cannot be reached due to previous paths are surrounded by blue.

Example

a(6) = 22 as from a(5) = 9 the squares to which the knight could move that have not been previously visited are 4, 12, 22, 26, 28, 46, 48. Of these, 4 is blocked by the path between 3 and 6, while 12 is blocked by the path between 10 and 3. This leaves 22 as the smallest unvisited square that is not blocked by any previous path.

Crossrefs

Cf. A394364 (end squares starting at n), A394365 (path lengths starting at n), A316667, A394387, A336208, A383185.

Sequence in context: A358150 A335214 A394387 * A338288 A330008 A335844

Adjacent sequences: A394360 A394361 A394362 * A394364 A394365 A394366

Keyword

nonn,fini,full,walk

Author

Scott R. Shannon, Mar 18 2026

Status

approved