cp's OEIS Frontend

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).

%I A157795 #14 Apr 15 2025 13:04:18
%S A157795 1,2,4,6,9,12,15,18,22,26,31,35,40
%N 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).
%C A157795 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.
%H A157795 Polymath1 wiki, <a href="http://michaelnielsen.org/polymath1/index.php?title=Fujimura%27s_problem">Fujimura's problem</a>
%H A157795 Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, <a href="https://doi.org/10.1609/aaai.v39i2.32162">Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences</a>, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
%e A157795 For n=2, a four-point set without triangles is (2,0,0), (0,0,2), (1,1,0), (0,1,1).
%K A157795 hard,more,nonn
%O A157795 0,2
%A A157795 _Terence Tao_, Mar 07 2009