A394388 The final square visited by a (1,2)-darter moving on a square-spiral numbered board, starting from square n, and always moving to the lowest available unvisited square that can be reached without crossing a square that has been previously visited (Cf. A394387).

90, 58, 29, 31, 34, 80, 49, 7, 36, 20, 3, 90, 102, 15, 30, 28, 36, 11, 137, 26, 50, 25, 6, 11, 43, 36, 19, 21, 19, 22, 34, 21, 15, 22, 20, 17, 8, 46, 41, 21, 80, 24, 26, 21, 4, 4, 4, 23, 30, 48, 73, 3, 98, 3, 24, 42, 22, 200, 15, 19, 15, 16, 15, 81, 17, 48, 11, 42

Offset

1,1

Comments

There are 61 possible squares that all walks can end on, these being 3, 4, 6, 7, 8, ..., 137, 138, 145, 200. The first path to end on square 3, the smallest possible value, is that starting on square 11, while he first, and only, path to end on square 200, the largest possible value, is that starting on square 58, which ends after visiting 121 squares. See the attached images.

See A394387 for more information, and A394389 for the total number of visited squares when starting from square n.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image of the path for a(11) = 3. This is the smallest possible end square. The start and end squares are colored green and red respectively, while the 8 squares around the final square that the darter is blocked from moving to are colored blue.

Scott R. Shannon, Image of the path for a(58) = 200. This is the largest possible end square.

Example

a(1) = 90. See A394387.
a(11) = 3. See the attached image.
a(58) = 200. See the attached image.

Crossrefs

Cf. A394387, A394389 (path lengths), A316667, A394363, A336208, A383185.

Sequence in context: A332427 A033410 A008901 * A161823 A270173 A003899

Adjacent sequences: A394385 A394386 A394387 * A394389 A394390 A394391

Keyword

nonn,walk

Author

Scott R. Shannon, Mar 19 2026

Status

approved