A393473 Consider the square spiral in the first quadrant, as in A393433, and look at the trajectory of a knight, starting at square n, and always moving to the smallest unvisited square. Sequence gives the total length of the trajectory.

53, 44, 111, 64, 53, 64, 177, 48, 53, 349, 112, 44, 47, 42, 177, 40, 98, 140, 46, 44, 74, 80, 416, 222, 125, 119, 207, 42, 43, 520, 45, 48, 43, 52, 16, 133, 187, 193, 80, 56, 41, 103, 666, 73, 509, 490, 53, 38, 222, 349, 136, 62, 43, 44, 532, 46, 102, 46, 49, 46

Offset

0,1

Comments

See A393472 for further details.

For starting squares less than 10000 (for larger values the initial knights path back toward the origin starts to dominate the total trajectory length) the longest trajectory is 666 for start square 42, while the shortest is 16 for start square 34. See the attached images.

Links

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Scott R. Shannon, The knight's path for a(34) = 16. In this and the other image the starting square is green, the last square is red, and the squares surrounding the last square that block the knight are surrounded by blue.

Scott R. Shannon, The knight's path for a(42) = 666.

Example

a(34) = 16 as when starting at square 34 the complete knight's path is 34 -> 13 -> 2 -> 9 -> 6 -> 1 -> 4 -> 7 -> 0 -> 5 -> 8 -> 3 -> 14 -> 11 -> 18 -> 15, visiting 16 squares in total. At square 15 the sequence ends as from 15 all available squares have been previously visited.

Crossrefs

Cf. A393433, A393472 (last square of trajectory), A316588, A307422, A316328.

Sequence in context: A171132 A259091 A104936 * A262908 A109648 A109733

Adjacent sequences: A393470 A393471 A393472 * A393474 A393475 A393476

Keyword

nonn

Author

Scott R. Shannon, Feb 15 2026

Status

approved