A358278 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 is on a different square ring of numbers than the current square.

1, 10, 3, 16, 33, 4, 11, 8, 19, 38, 5, 14, 29, 2, 13, 28, 9, 12, 27, 24, 7, 18, 35, 60, 15, 6, 17, 34, 59, 30, 53, 26, 79, 46, 21, 40, 67, 36, 61, 32, 55, 86, 51, 48, 23, 44, 71, 20, 39, 66, 99, 62, 37, 68, 41, 22, 43, 70, 105, 148, 65, 98, 139, 94, 31, 54, 85, 50, 25, 52, 49, 78, 45, 74

Offset: 1

Author

Scott R. Shannon and Eric Angelini, Nov 08 2022

Keywords

• nonn
• fini

Comments

This sequence is finite: after 1455 squares have been visited the square with number 1345 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(1374) = 1996 while the smallest unvisited square is 1024.

Examples

			The board is numbered using a square spiral. The square rings of numbers are shown below:
.
    17--16--15--14--13   .
     |               |   .
    18   5---4---3  12  29
     |   |       |   |   |
    19   6   1   2  11  28
     |   |       |   |   |
    20   7---8---9  10  27
     |               |   |
    21--22--23--24--25  26
                         |
   -44--45--46--47--48--49
.
a(4) = 16 as after the knight moves to the square containing a(3) = 3 the available unvisited squares are 6, 8, 16, 28, 30, 32, 34. Of these 6 and 8 are the smallest but both of them lie on the first square ring of numbers, the same as the current number 3. Of the remaining squares the smallest unvisited square is 16. This is the first term to differ from A316667.
		

Links

• Scott R. Shannon, Image showing the knight's path on the square spiral. The starting 1 square is shown as a green dot while the final square numbered 1345, near the middle of the bottom edge, is shown as a red dot. Also shown as blue dots are the eight occupied squares around the final square.

Crossrefs

Cf. A316667, A358150, A328929, A328909, A174344, A274923.