A328928 Squares visited by a knight moving on a taxicab geometry numbered board where the knight moves to the smallest numbered unvisited square; the minimum distance from the origin is used if the square numbers are equal; the smallest spiral number ordering is used if the distances are equal.

0, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 5, 2, 3, 4, 3, 4, 3, 2, 5, 4, 3, 4, 5, 2, 3, 4, 5, 4, 5, 4, 3, 4, 5, 6, 3, 4, 3, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 6, 5, 4, 5, 4, 7, 6, 5, 6, 7, 6, 5, 6, 5, 4, 7, 6, 5, 6, 7, 6, 5, 8, 7, 6, 7, 6, 5, 6, 7, 6, 5, 6, 5, 6, 7, 8, 7, 8, 7, 6, 7, 8, 7, 6, 7, 8, 7, 6

Offset: 0

Author

Scott R. Shannon, Oct 31 2019

Keywords

• nonn
• fini

Comments

This sequence uses the taxicab geometry distance from the 0-squared origin to enumerate each square on the board. At each step the knight goes to an unvisited square with the smallest square number. If the knight has a choice of two or more squares with the same number it then chooses the square which is the closest to the 0-squared origin. If two or more squares are found which also have the same distance to the origin, then the square which was first drawn in a square spiral numbering is chosen, i.e., the smallest spiral numbered square as in A316667.

The sequence is finite. After 1092366 steps a square with the number 728 (standard spiral number = 1165673) is visited, after which all neighboring squares have been visited.

See A328908(n) for the position on the spiral (cf. A316328) of the square visited at move n. - M. F. Hasler, Nov 04 2019

Examples

			The squares are labeled using their taxicab geometry distance from the origin:
.
    +----+----+----+----+----+----+----+
    |  6 |  5 |  4 |  3 |  4 |  5 |  6 |
    +----+----+----+----+----+----+----+
    |  5 |  4 |  3 |  2 |  3 |  4 |  5 |
    +----+----+----+----+----+----+----+
    |  4 |  3 |  2 |  1 |  2 |  3 |  4 |
    +----+----+----+----+----+----+----+
    |  3 |  2 |  1 |  0 |  1 |  2 |  3 |
    +----+----+----+----+----+----+----+
    |  4 |  3 |  2 |  1 |  2 |  3 |  4 |
    +----+----+----+----+----+----+----+
    |  5 |  4 |  3 |  2 |  3 |  4 |  5 |
    +----+----+----+----+----+----+----+
    |  6 |  5 |  4 |  3 |  4 |  5 |  6 |
    +----+----+----+----+----+----+----+
.
If the knight has a choice of two or more squares with the same number which also have the same distance from the origin, then the square with the minimum square spiral number, as shown in A316667, is chosen.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 0..20000
• M. F. Hasler, Knight tours, OEIS wiki, Nov. 2019
• Scott R. Shannon, Image showing the 1092366 steps of the knight's path. The green dot is the first square and the red dot the last. Blue dots show the eight occupied squares surrounding the final square; the final square is on the boundary at about the 12:30 clock position.
• N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).
• Wikipedia, Taxicab geometry.

Crossrefs

Cf. A328908, A316667, A326922.

Cf. A174344, A274923, A296030 (coordinates of the n-th point on the spiral).

Programs

PARI

A328928(n)=normlp(coords(A328908(n)),1) \\ with coords() defined e.g. in A296030. - M. F. Hasler, Nov 04 2019

Formula

a(n) = |A174344(p)| + |A274923(p)| with p = A328908(n)+1. - M. F. Hasler, Nov 04 2019