A357234 a(n) is the maximum length of a snake-like polyomino in an n X n square that starts and ends at opposite corners.

1, 3, 5, 7, 17, 23, 31, 39, 51, 63, 75, 89, 105, 121, 139, 159

Offset

1,2

Comments

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).

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?

Links

Table of n, a(n) for n=1..16.

Yi Yang, The longest snakes presently known that start and end at opposite corners in the n X n square up to n = 17.

Yi Yang, A post that shows the lower bounds for a(18)-a(22).

Yi Yang, A C++ program that generates upper bounds for a(n) up to n = 19.

Formula

a(n) ~ 2*n^2/3. - Pontus von Brömssen, Sep 19 2022

a(n) <= A331968(n). - Pontus von Brömssen, Sep 21 2022

Example

Longest snakes for 5 <= n <= 8:
  X X X X X   X X X X X X   X X X . X X X   X . X X X X X X
  . . . . X   . . . . . X   . . X . X . X   X . X . . . . X
  X X X X X   X X X X X X   X X X . X . X   X . X X X X . X
  X . . . .   X . . . . .   X . . X X . X   X X . . . X . X
  X X X X X   X . X X X X   X . . X . X X   . X . X X X . X
              X X X . . X   X . . X . X .   X X . X . . X X
                            X X X X . X X   X . . X . . X .
                                            X X X X . . X X

Crossrefs

Cf. A331968, A357516.

Sequence in context: A137258 A053341 A331894 * A086086 A141772 A220454

Adjacent sequences: A357231 A357232 A357233 * A357235 A357236 A357237

Keyword

nonn,hard,more

Author

Yi Yang, Sep 18 2022

Extensions

a(1)-a(9) confirmed by Pontus von Brömssen, Sep 21 2022. - N. J. A. Sloane, Sep 30 2022

a(10)-a(13) confirmed by Elijah Beregovsky, Nov 27 2022

a(14)-a(16) from Andrew Howroyd, Feb 28 2023

Status

approved