There are many related sequences in OEIS: https://oeis.org/ try author:Hardin (33890 results) or better no three in line (123 results)
Вторник, 10 мая 2016, 20:00 +03:00 от Warren D Smith <warren.wds@gmail.com>:
Good programming-contest problem: Let F(N) be the max possible number of points you can pick from an NxN grid, such that no 3 lie on a line. Find the best lower bounds you can for F(1), F(2), ..., F(256).
I have a program which appears to be capable of showing F(N)>1.57*N for an infinite set of N, although the 1.57 is just a guess based on looking at its results for various N, and might be deluded. (The fact that 1.57 approximates pi/2 is probably just a coincidence?) My program provably exceeds 1.4999*N for an infinite set of N.
A trivial upper bound is F(N)<=2*N, because if you pick 2*N+1 points, then 3 must lie on some horizontal gridline. This trivial upper bound has actually been attained for N=52 and also most N below 52. That was done by Achim Flammenkamp. Perhaps 2N is also attainable for some N>52, perhaps even an infinite set of N. That is unknown.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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