OFFSET
1,2
COMMENTS
The triangle is oriented with apex at the top and horizontal base.
Label the entries in the top left and right edges with the numbers 1 through 2n-1, and let S denote the subset of [1..2n-1] where these edges contains 1's. Then the matrix has the no-subtriangle property iff S contains no three-term arithmetic progression.
EXAMPLE
n, a(n), example of optimal S:
1, 1, [1]
2, 2, [1, 2]
3, 4, [1, 3, 4]
4, 6, [1, 2, 4, 5]
5, 8, [2, 3, 5, 6]
6, 10, [3, 4, 6, 7]
7, 13, [1, 5, 7, 8, 10]
8, 16, [1, 2, 7, 8, 10, 11]
9, 20, [1, 3, 4, 9, 10, 12, 13]
10, 24, [1, 2, 4, 5, 10, 11, 13, 14]
11, 28, [2, 3, 5, 6, 11, 12, 14, 15]
12, 32, [3, 4, 6, 7, 12, 13, 15, 16]
13, 36, [4, 5, 7, 8, 13, 14, 16, 17]
14, 40, [5, 6, 8, 9, 14, 15, 17, 18]
...
For example, the line 5, 8, [2, 3, 5, 6] corresponds to the triangle
....1....
...0.1...
..1.1.0..
.1.0.1.0.
0.1.1.0.0
and the value a(5) = 8.
It is a plausible conjecture that any optimal solution S here is also an optimal solution to the square grid version in A269745, and vice versa. (The square grid being obtained by reflecting the triangle in its base.)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Warren D. Smith and N. J. A. Sloane, Mar 20 2016
STATUS
approved