A157795 Largest subset of the discrete triangular grid { (a,b,c): a+b+c = n, a,b,c >= 0 } that does not contain any upward-pointing triangles (i.e., triples (a+r,b,c), (a,b+r,c), (a,b,c+r) with r positive).

1, 2, 4, 6, 9, 12, 15, 18, 22, 26, 31, 35, 40

Offset: 0

Author

Terence Tao, Mar 07 2009

Keywords

• hard
• more
• nonn

Comments

The n=3 case was posed as a problem by Fujimura. The sequence is related to a certain "hyper-optimistic conjecture" regarding the density Hales-Jewett theorem.

Examples

			For n=2, a four-point set without triangles is (2,0,0), (0,0,2), (1,1,0), (0,1,1).
		

Links

• Polymath1 wiki, Fujimura's problem
• Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.