A393318 Final square visited by a knight moving on a square-spiral numbered board when starting from square n and always moving to the lowest available unvisited square.

2084, 711, 3915, 556, 3915, 556, 3915, 3380, 2086, 1339, 1464, 1572, 4772, 582, 3959, 682, 2309, 385, 330, 1142, 706, 2750, 4256, 4322, 1413, 1488, 1685, 6335, 5214, 906, 2616, 1038, 582, 1608, 2675, 6266, 1120, 3955, 936, 2773, 1861, 2213, 2223, 262, 1147, 4428

Offset

1,1

Comments

There are 1518 distinct values, see A306291, the last one being a(17390) = 16851 beyond which the terms enter a repeating cycle of values as the knight starts further and further away from the origin.

See A316667 and A306291 for further details and images of the paths.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image of the first 50000 terms.

N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Example

a(1) = 2084. See A316667.
a(175) = 104. This is the smallest possible final square number. See A306291.
a(11509) = 23134. This is the largest possible final square number. See A306291.

Crossrefs

Cf. A306291, A316667, A307422, A393472.

Sequence in context: A323813 A323472 A224438 * A323714 A343179 A385003

Adjacent sequences: A393315 A393316 A393317 * A393319 A393320 A393321

Keyword

nonn,walk

Author

Scott R. Shannon, Mar 11 2026

Status

approved