A350547: Equilateral Triangle-Free 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.

  1. n = 0

    a(0) = 1

  2. n = 1

    a(1) = 4

  3. n = 2

    a(2) = 9

  4. n = 3

    a(3) = 15

  5. n = 4

    a(4) = 22

  6. n = 5

    a(5) = 28

  7. n = 6

    a(6) = 36

  8. n = 7

    a(7) ≥ 44

  9. n = 8

    a(8) ≥ 52

  10. n = 9

    a(9) ≥ 60

  11. n = 10

    a(10) ≥ 66

  12. n = 11

    a(11) ≥ 74

  13. n = 12

    a(12) ≥ 82

  14. 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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