A394387 - OEIS

%I #11 Mar 23 2026 08:52:29

%S 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,

%T 21,70,39,36,61,16,63,38,65,98,35,96,33,58,55,28,51,48,77,24,79,26,81,

%U 84,49,82,119,78,115,74,109,112,155,108,69,104,37,68,103,100

%N 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.

%C 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.

%C 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.

%H Scott R. Shannon, <a href="/A394387/b394387.txt">Table of n, a(n) for n = 1..88</a>

%H Scott R. Shannon, <a href="/A394387/a394387.png">Image of the full path</a>. 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.

%e 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.

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

%K nonn,fini,full,walk

%O 1,2

%A _Scott R. Shannon_, Mar 19 2026