A357234 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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