A157795 - OEIS

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 #10 Jan 19 2019 04:14:58

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

%N 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 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 Polymath1 wiki, <a href="http://michaelnielsen.org/polymath1/index.php?title=Fujimura%27s_problem">Fujimura's problem</a>

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

%K hard,more,nonn

%O 0,2

%A _Terence Tao_, Mar 07 2009