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%I #9 Nov 01 2022 07:50:28
%S 1,10,3,6,9,12,15,18,35,14,11,24,27,48,23,20,39,36,61,32,29,52,25,28,
%T 51,80,47,76,43,70,105,38,63,34,59,56,87,126,53,84,49,78,45,74,71,106,
%U 67,64,97,60,93,90,55,58,89,92,131,88,127,174,83,120,79,116,75,72,107,68,103,100,141
%N Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the smallest numbered unvisited square and where the square number is more than the number of currently visited squares.
%C This sequence is finite: after 15767 squares have been visited the square with number 15813 is reached after which all eight neighboring squares the knight could move to have already been visited. See the linked image. The largest visited square is a(15525) = 19363, while numerous smaller numbered squares are never visited, e.g., 2, 4, 5, 7, 8, 13, 16, 17, 19, ... .
%H Scott R. Shannon, <a href="/A358150/a358150.png">Image showing the knight's path on the square spiral</a>. The starting 1 square is shown as a green dot while the final square numbered 15813, near the middle of the top edge, is shown as a red dot. Also shown as blue dots are the eight occupied squares around the final square.
%e The board is numbered with the square spiral:
%e .
%e 17--16--15--14--13 .
%e | | .
%e 18 5---4---3 12 29
%e | | | | |
%e 19 6 1---2 11 28
%e | | | |
%e 20 7---8---9--10 27
%e | |
%e 21--22--23--24--25--26
%e .
%e a(6) = 12 as after the knight moves to the square containing 9 the available unvisited squares are 4, 12, 22, 26, 28, 46, 48. Of these 4 is the smallest but as we have already visited five squares that cannot be chosen. Of the remaining squares greater than five the smallest unvisited square is 12. This is the first term to differ from A316667.
%Y Cf. A316667, A326918, A326922, A316588.
%K nonn,fini,walk
%O 1,2
%A _Scott R. Shannon_, Nov 01 2022
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