A331968 Maximum number of unit squares of a snake-like polyomino in an n X n square box.

1, 3, 7, 11, 17, 24, 33, 42, 53, 64, 77, 92, 107, 123, 142, 162, 182

Offset: 1

Author

Alain Goupil, Feb 02 2020

Keywords

• nonn
• hard
• more

Comments

These are similar to the snake-in-the-box problem for the hypercube Q_n (See A099155).

The number of solutions is given by A331986(n).

Equivalently, a(n) is the maximum number of vertices in a path without chords in the n X n grid graph. A path without chords is an induced subgraph that is a path.

These numbers are part of the result of a computer program that counts the snake-like polyominoes in a rectangle of given size b X h by their length.

a(16) >= 161.

Examples

			For n=4, the maximum length of a snake-like polyomino that fits in a square of side 4 is 11 and there are 84 such snakes.
Maximum-length snakes for n = 1 to 4 are shown below.
   X    X X    X X X    X X X X
        X      X   X    X     X
               X   X    X     X
                        X   X X
		

Links

• Nikolai Beluhov, Snake paths in king and knight graphs, arXiv:2301.01152 [math.CO], 2023.
• Alain Goupil, Illustration of initial terms
• Eric Weisstein's World of Mathematics, Grid Graph

Crossrefs

Main diagonal of A360917.

Cf. A099155, A047838, A122224, A331986, A332920, A332921, A357357, A357359.

Formula

a(n) >= A047838(n+1).

For n > 2: a(n) >= 2*floor(n/3)*(2n-3*floor(n/3)-2)+5. - Elijah Beregovsky, May 11 2020

a(n) <= (2*n*(n+1)-1)/3. - Elijah Beregovsky, Nov 09 2020

a(n) = 2*n^2/3 + O(n) (Beluhov 2023). - Pontus von Brömssen, Jan 30 2023

Extensions

a(15) from Andrew Howroyd, Feb 04 2020

a(16)-a(17) from Yi Yang, Oct 03 2022