OFFSET
0,3
COMMENTS
The hexagonal grid of size n, or side length n, consists of 2n-1 centered rows of respective length (number of cells or available grid points) n, n+1, ..., 2n-1, 2n-2, ..., n, for a total of A003215(n-1) = 3n(n-1) + 1 cells, cf. examples.
The forbidden arithmetic progressions can also be stated by requiring that all 3-element subsets must be non-averaging, i.e., none of the three points can be the center or midpoint of the two other points.
Lower bounds for the next terms are a(8) >= 52, a(9) >= 59, a(10) >= 68, a(11) >= 80. - M. A. Achterberg, Feb 10 2022
LINKS
Al Zimmermann, Programming Contest: AP Math, Sep. - Dec. 2021
EXAMPLE
For n = 2, the best solution is to leave out only the central point, which yields a(2) = 6, see the drawing below. If the central point is used, for any of the peripheral points, the opposite one must remain empty, so the maximum score is only 1 + 3 = 4.
n = 0: n = 1: n = 2: n = 3: X X O n = 4: X O X X
X X X X O X X X O X X
[] X X O X O O O X X O X O O O O
X X X X O X X O O O O O X
X X O O O O O X O
n = 5: X X O X X X X O X X
X X O O X X X X O X
O O O O O O O
X O X O O X O X
X X O O O O O X X
X O O O O O O X
O O O X O O O
X X O O X X
X X O X X
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
M. F. Hasler, Jan 11 2022
EXTENSIONS
a(7) from M. A. Achterberg, Feb 10 2022
STATUS
approved