login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
LINKS
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.
FORMULA
a(n) = |A174344(p)| + |A274923(p)| with p = A328908(n)+1. - M. F. Hasler, Nov 04 2019
EXAMPLE
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.
PROG
(PARI) A328928(n)=normlp(coords(A328908(n)), 1) \\ with coords() defined e.g. in A296030. - M. F. Hasler, Nov 04 2019
CROSSREFS
Cf. A174344, A274923, A296030 (coordinates of the n-th point on the spiral).
Sequence in context: A077089 A156352 A175191 * A370560 A342468 A324534
KEYWORD
nonn,fini
AUTHOR
Scott R. Shannon, Oct 31 2019
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 13 14:24 EDT 2024. Contains 374284 sequences. (Running on oeis4.)