A390110 Number of loops with length A389534(n) touching all cells in an n X n grid.

1, 1, 1, 8, 2, 8, 6, 24, 128

Offset

2,4

Comments

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

"Touch" includes vertices and edges of grid cells.

Using the shortest known lengths: a(11) to a(13) >= 192, 12, 120 and a(6*k) >= 4^k/2 + 2^k for k >= 2, with equality iff k = 2. [Updated by Tamás Fülöp, Dec 14 2025]

Links

Table of n, a(n) for n=2..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.

Formula

a(0) = 1.

a(1) = oo.

a(n) <= 8 * A390109(n), if n != 1, with equality for all odd n >= 5.

Crossrefs

Cf. A389534, A390109.

Sequence in context: A246849 A143531 A352473 * A211269 A278809 A381672

Adjacent sequences: A390107 A390108 A390109 * A390111 A390112 A390113

Keyword

nonn,more,hard

Author

Tamás Fülöp, Oct 25 2025

Status

approved