A329022 -id:A329022 - cp's OEIS frontend

A316667 Squares visited by a knight moving on a spirally numbered board always to the lowest available unvisited square.

1, 10, 3, 6, 9, 4, 7, 2, 5, 8, 11, 14, 29, 32, 15, 12, 27, 24, 45, 20, 23, 44, 41, 18, 35, 38, 19, 16, 33, 30, 53, 26, 47, 22, 43, 70, 21, 40, 17, 34, 13, 28, 25, 46, 75, 42, 69, 104, 37, 62, 95, 58, 55, 86, 51, 48, 77, 114, 73, 108, 151, 68, 103, 64, 67, 36

Offset: 1

Daniël Karssen, Jul 10 2018, following a suggestion from N. J. A. Sloane, Jul 09 2018

Board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   .
   |   |       |   |   .
  19   6   1---2  11   .
   |   |           |   .
  20   7---8---9--10   .
   |                   .
  21--22--23--24--25--26
.
This sequence is finite: At step 2016, square 2084 is visited, after which there are no unvisited squares within one knight move.

Daniël Karssen, Table of n, a(n) for n = 1..2016
Daniël Karssen, Figure showing the first 60 steps of the sequence
Daniël Karssen, Figure showing the complete sequence
Daniël Karssen, MATLAB script to generate the complete sequence
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019)

Cf. A316328 (same starting at 0), A329022 (same with diamond-shaped spiral), A316588 (variant on board with x,y >= 0).
Cf. A326924 (choose square closest to the origin), A328908 (using taxicab distance), A328909 (using sup norm); A323808, A323809.
The (x,y) coordinates of square k are (A174344(k), A274923(k)).

A316667(n)=A316328(n-1)+1 \\ M. F. Hasler, Nov 06 2019

a(n) = A316328(n-1) + 1.

A332837 Squares visited by a knight moving on a double spiral numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 10, 5, 2, 8, 7, 4, 3, 9, 6, 12, 18, 33, 39, 20, 11, 32, 19, 13, 22, 28, 15, 21, 38, 61, 30, 17, 42, 25, 31, 16, 43, 24, 51, 76, 26, 45, 70, 37, 14, 29, 23, 40, 34, 57, 86, 49, 55, 84, 78, 53, 47, 72, 107, 41, 35, 56, 27, 44, 71, 36, 59, 88, 127, 80, 115

Offset: 1

Scott R. Shannon, Feb 26 2020

This sequence uses a double spiral of numbers to enumerate the squares on the board. The knight starts on the square with number 1. At each step the knight goes to an unvisited square with the smallest number.
The sequence is finite. After 2958 steps the square with number 2796 is visited, after which all neighboring squares have been visited.
The lowest unvisited square during the walk is square number 2011.

			The squares are numbered using the double spiral numbering shown below:
.
  --48--46--44--42--40--38--36
                             |
    27--25--23--21--19--17  34
     |                   |   |
    29  10---8---6---4  15  32
     |   |           |   |   |
    31  12   3---1---2  13  30
     |   |   |           |   |
    33  14   5---7---9--11  28
     |   |                   |
    35  16--18--20--22--24--26
     |
    37--39--41--43--45--47--49--

Scott R. Shannon, Table of n, a(n) for n = 1..2959
Scott R. Shannon, Image showing the 2958 steps of the knights' path. The green dot is the starting square and the red dot the final square. Blue dots show the eight occupied squares surrounding the final square. The lowest unvisited square is the yellow dot.

Cf. A220098, A316667, A329022, A332980 (quadruple spiral).

A332980 Squares visited by a knight moving on a quadruple spiral numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 10, 7, 2, 8, 3, 9, 4, 6, 5, 15, 12, 38, 11, 14, 21, 32, 20, 17, 28, 39, 13, 16, 19, 30, 18, 33, 44, 56, 37, 22, 34, 23, 35, 24, 36, 25, 52, 71, 29, 40, 59, 47, 66, 31, 42, 54, 77, 89, 27, 62, 43, 55, 74, 86, 117, 70, 51, 94, 67, 48, 60, 41, 26, 65, 92, 49

Offset: 1

Scott R. Shannon, Mar 04 2020

This sequence uses a quadruple spiral of numbers to enumerate the squares on the board. The knight starts on the square with number 1. At each step the knight goes to an unvisited square with the smallest number.
The sequence is finite. After 1837 steps the square with number 1748 is visited, after which all neighboring squares have been visited.
The lowest unvisited square during the walk is square number 1211.

			The squares are numbered using the quadruple spiral numbering shown below:
                             |
  --49--45--41--37--33--29  48
                         |   |
    26--22--18--14--10  25  44
     |               |   |   |
    30  11---7---3   6  21  40
     |   |       |   |   |   |
    34  15   4-- 1---2  17  36
     |   |   |   |       |   |
    38  19   8   5---9--13  32
     |   |   |               |
    42  23  12--16--20--24--28
     |   |
    46  27--31--35--39--43--47--
     |

Scott R. Shannon, Table of n, a(n) for n = 1..1838
Scott R. Shannon, Image showing the 1837 steps of the knights' path. The green dot is the starting square and the red dot the final square. Blue dots show the eight occupied squares surrounding the final square. The lowest unvisited square is the yellow dot.

Cf. A220098, A316667, A329022, A332837 (double spiral).

cp's OEIS Frontend

A316667 Squares visited by a knight moving on a spirally numbered board always to the lowest available unvisited square.

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A332837 Squares visited by a knight moving on a double spiral numbered board and moving to the lowest available unvisited square at each step.

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A332980 Squares visited by a knight moving on a quadruple spiral numbered board and moving to the lowest available unvisited square at each step.

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