A390109 Number of canonical representative loops with length A389534(n) touching all cells in an n X n grid.

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 18

Offset

0,9

Comments

"Loop" means any continuous curve starting and ending at the same point.

"Touch" includes vertices and edges of grid cells.

"Canonical representative" is a single loop chosen from its orbit under the Euclidean group E(2).

Using the shortest known lengths: a(11) to a(13) >= 24, 3, 15 and a(6k) >= 4^(k-2)+2^(k-2) + binomial(k-1,floor((k-1)/2)), for k >= 2, with equality iff k = 2.

Links

Table of n, a(n) for n=0..10.

Zoe Allen, Touching as many grid squares as possible with path of given length, Math StackExchange, Feb 18 2025.

Tamás Fülöp, C++ program, GitHub repository, Oct 07 2025.

Tamás Fülöp, All loops with n = 3 - 10, Oct 07 2025.

Crossrefs

Cf. A389534.

Sequence in context: A394419 A132534 A263048 * A118702 A073524 A130226

Adjacent sequences: A390106 A390107 A390108 * A390110 A390111 A390112

Keyword

nonn,more,hard

Author

Tamás Fülöp, Oct 25 2025

Status

approved