A394388 - OEIS

%I #12 Mar 23 2026 08:52:25

%S 90,58,29,31,34,80,49,7,36,20,3,90,102,15,30,28,36,11,137,26,50,25,6,

%T 11,43,36,19,21,19,22,34,21,15,22,20,17,8,46,41,21,80,24,26,21,4,4,4,

%U 23,30,48,73,3,98,3,24,42,22,200,15,19,15,16,15,81,17,48,11,42

%N 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).

%C 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.

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

%H Scott R. Shannon, <a href="/A394388/b394388.txt">Table of n, a(n) for n = 1..10000</a>

%H Scott R. Shannon, <a href="/A394388/a394388.png">Image of the path for a(11) = 3</a>. 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.

%H Scott R. Shannon, <a href="/A394388/a394388_1.png">Image of the path for a(58) = 200</a>. This is the largest possible end square.

%e a(1) = 90. See A394387.

%e a(11) = 3. See the attached image.

%e a(58) = 200. See the attached image.

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

%K nonn,walk

%O 1,1

%A _Scott R. Shannon_, Mar 19 2026