%I #48 Feb 28 2023 13:07:15
%S 1,3,5,7,17,23,31,39,51,63,75,89,105,121,139,159
%N a(n) is the maximum length of a snake-like polyomino in an n X n square that starts and ends at opposite corners.
%C Snake-like polyominoes have all cells with at most two neighbor cells, and have at least one cell that has only one neighbor cell, where neighbors are horizontal or vertical (not diagonal).
%C Lower bounds for a(10)-a(22) are 63, 75, 89, 105, 121, 139, 159, 179, 201, 225, 249, 275, 303. Is it true that a(n) = round((2*n*n-4*n+28)/3) for n >= 9?
%H Yi Yang, <a href="/A357234/a357234.png">The longest snakes presently known that start and end at opposite corners in the n X n square up to n = 17</a>.
%H Yi Yang, <a href="https://tieba.baidu.com/p/8049203368">A post that shows the lower bounds for a(18)-a(22)</a>.
%H Yi Yang, <a href="https://bbs.emath.ac.cn/thread-18542-4-1.html">A C++ program that generates upper bounds for a(n) up to n = 19</a>.
%F a(n) ~ 2*n^2/3. - _Pontus von Brömssen_, Sep 19 2022
%F a(n) <= A331968(n). - _Pontus von Brömssen_, Sep 21 2022
%e Longest snakes for 5 <= n <= 8:
%e X X X X X X X X X X X X X X . X X X X . X X X X X X
%e . . . . X . . . . . X . . X . X . X X . X . . . . X
%e X X X X X X X X X X X X X X . X . X X . X X X X . X
%e X . . . . X . . . . . X . . X X . X X X . . . X . X
%e X X X X X X . X X X X X . . X . X X . X . X X X . X
%e X X X . . X X . . X . X . X X . X . . X X
%e X X X X . X X X . . X . . X .
%e X X X X . . X X
%Y Cf. A331968, A357516.
%K nonn,hard,more
%O 1,2
%A _Yi Yang_, Sep 18 2022
%E a(1)-a(9) confirmed by _Pontus von Brömssen_, Sep 21 2022. - _N. J. A. Sloane_, Sep 30 2022
%E a(10)-a(13) confirmed by _Elijah Beregovsky_, Nov 27 2022
%E a(14)-a(16) from _Andrew Howroyd_, Feb 28 2023