%I #86 Feb 27 2023 11:16:21

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

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

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

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

%C 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.

%C 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.

%C a(16) >= 161.

%H Nikolai Beluhov, <a href="https://arxiv.org/abs/2301.01152">Snake paths in king and knight graphs</a>, arXiv:2301.01152 [math.CO], 2023.

%H Alain Goupil, <a href="/A331968/a331968_2.pdf">Illustration of initial terms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>

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

%F For n > 2: a(n) >= 2*floor(n/3)*(2n-3*floor(n/3)-2)+5. - _Elijah Beregovsky_, May 11 2020

%F a(n) <= (2*n*(n+1)-1)/3. - _Elijah Beregovsky_, Nov 09 2020

%F a(n) = 2*n^2/3 + O(n) (Beluhov 2023). - _Pontus von Brömssen_, Jan 30 2023

%e 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.

%e Maximum-length snakes for n = 1 to 4 are shown below.

%e X X X X X X X X X X

%e X X X X X

%e X X X X

%e X X X

%Y Main diagonal of A360917.

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

%K nonn,hard,more

%O 1,2

%A _Alain Goupil_, Feb 02 2020

%E a(15) from _Andrew Howroyd_, Feb 04 2020

%E a(16)-a(17) from _Yi Yang_, Oct 03 2022