A394387 Squares visited by a (1,2)-darter moving on a square-spiral numbered board, starting from square 1, always moving to the lowest available unvisited square that can be reached without crossing a square that has been previously visited.

1, 10, 3, 6, 9, 22, 19, 4, 13, 2, 27, 12, 31, 14, 5, 34, 17, 40, 7, 42, 45, 8, 25, 46, 75, 72, 21, 70, 39, 36, 61, 16, 63, 38, 65, 98, 35, 96, 33, 58, 55, 28, 51, 48, 77, 24, 79, 26, 81, 84, 49, 82, 119, 78, 115, 74, 109, 112, 155, 108, 69, 104, 37, 68, 103, 100

Offset

1,2

Comments

A (1,2)-darter can move to any of eight surrounding squares reachable by a standard knight, but to get there is must slide along either of the two available L-shaped paths, and if any square along a given L-shaped path has been previously visited then that L-shaped path is blocked.

The path is finite - after visiting 88 squares, the square with number 90 is reached, after which all 8 squares the darter could move to have either been previously visited or are blocked by visited squares of previous steps. See the attached image.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..88

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 visited squares blocking both L-shaped paths, are surrounded by blue.

Example

a(6) = 22 as from a(5) = 9 the squares to which the darter could move that have not been previously visited are 4, 12, 22, 26, 28, 46, 48. Of these, 4 has both L-shaped paths to it blocked by visited squares 1 and 3, while 12 has both L-shaped paths to it blocked by squares 3 and 10. This leaves 22 as the smallest unvisited square that has at least one L-shaped path free of previously visited squares.

Crossrefs

Cf. A394388 (end squares starting at n), A394389 (path lengths starting at n), A316667, A394363, A336208, A383185.

Sequence in context: A362027 A358150 A335214 * A394363 A338288 A330008

Adjacent sequences: A394384 A394385 A394386 * A394388 A394389 A394390

Keyword

nonn,fini,full,walk

Author

Scott R. Shannon, Mar 19 2026

Status

approved