A350547: Equilateral TriangleFree Hexagon Lattices
Description
Let a(n) be the maximum size of a set of points taken from a hexagonal section of a hexagonal grid with side length n such that no three selected points form an equilateral triangle. What is a(n) for small values of n? How does a(n) grow as n increases?
Best Known Solutions
Here is a sample of known solutions and lower bounds. For many of these, there are additional symmetric and asymmetric solutions which are maximal.

n = 0
a(0) = 1

n = 1
a(1) = 4

n = 2
a(2) = 9

n = 3
a(3) = 15

n = 4
a(4) = 22

n = 5
a(5) = 28

n = 6
a(6) = 36

n = 7
a(7) ≥ 44

n = 8
a(8) ≥ 52

n = 9
a(9) ≥ 60

n = 10
a(10) ≥ 66

n = 11
a(11) ≥ 74

n = 12
a(12) ≥ 82

n = 13
a(13) ≥ 94
Additional Notes
Additional (significantly weaker) symmetric lower bounds include: 98, 106, 116, 126, 132, 136, and 142 for a(14) to a(20).
It is possible to use known values to compute trivial upper bounds:
a(n) ≤ h(n) a(n  1) / h(n  1) where h(n) = 3 n (n + 1) + 1 is the total number of points in a hexagonal section of side length n.
A path of selected points will never form a closed loop (proof omitted).
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