A326413 Successive squares visited by a knight on the single-digit square spiral, with ties resolved towards the left.

0, 0, 1, 0, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 3, 2, 2, 1, 3, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 1, 4, 3, 5, 6, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0

Offset: 1

Author

N. J. A. Sloane, Oct 17 2019

Keywords

• nonn
• fini
• full

Comments

Take the standard counterclockwise square spiral starting at 0, as in A304586, but only write one digit at a time in the cells of the spiral: 0,1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,...

Place a chess knight at cell 0. Move it to the lowest-numbered cell it can attack, and if there is a tie, move it to the cell closest (in Euclidean distance) to the start, and if there is still a tie, move to the left(*).

No cell can be visited more than once.

Inspired by the Trapped Knight video and A316667.

Just as for A316667, the sequence is finite. After a while, the knight has no unvisited squares it can reach, and the sequence ends with a(1217) = 4.

(*)Moving to the left means choose the point with the lowest x-coordinate. This leads to an unambiguous choice of tied squares only for the 'move left' case.

Examples

			The digit-square spiral is
                                .
                                .
    2---2---2---1---2---0---2   2
    |                       |   |
    3   1---2---1---1---1   9   3
    |   |               |   |   |
    2   3   4---3---2   0   1   1
    |   |   |       |   |   |   |
    4   1   5   0---1   1   8   3
    |   |   |           |   |   |
    2   4   6---7---8---9   1   0
    |   |                   |   |
    5   1---5---1---6---1---7   3
    |                           |
    2---6---2---7---2---8---2---9
		

Links

• Luca Petrone, Table of n, a(n) for n = 1..1217
• Eric Angelini, Kneil's Knumberphile Knight, Cinquante signes, May 04 2019.
• Eric Angelini, Kneil's Knumberphile Knight, Cinquante signes, May 04 2019. [Cached copy, pdf file, with permission]
• M. F. Hasler, Knight tours, OEIS wiki, Nov. 2019.
• N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Crossrefs

Cf. A304586, A316667, A328698.

Cf. A326916, A326918; A316328; A326924, A326922; A328908, A328928; A328909, A328929.

Extensions

More terms from Luca Petrone

Corrected and extended by Eric Angelini, Oct 24 2019