No 3 in line problem

[Ed Pegg Jr]

Actually, the last link added by Sloane is April 2006.

http://wso.williams.edu/~bchaffin/no_three_in_line/index.htm

 From my column on the topic:
http://www.maa.org/editorial/mathgames/mathgames_04_11_05.html

Richard Guy (pers comm, Oct 2004): "I got the no-three-in-line problem 
<http://wwwhomes.uni-bielefeld.de/achim/cgi/no3in/readme.html> from 
Heilbronn over 50 years ago. See F4 in UPINT 
<http://www.amazon.com/exec/obidos/tg/detail/-/0387208607/mathpuzzlecom/>. 
In Canad. Math. Bull. 11 (1968) 527--531; MR 39 #129 Guy & Kelly 
conjecture that, for large /n/, at most (/c/ + ?)/n/ points can be 
selected. Curiously, as recently as last March, Gabor Ellmann pointed 
out an error in our heuristic reasoning, which, when corrected, gives 
/c/ = ?/sqrt(3), or /c/ ~ 1.813799. I should send a correction to Canad 
Math Bull! For those with a lot of computer time to spare, there's a lot 
to be discovered. Apart from a limping odd-even phenomenon, the number 
of solutions with 2/n/ points appears to grow exponentially at first. 
Who will be the first to show that this begins to tail off? A000769 
<http://www.research.att.com/projects/OEIS?Anum=A000769> in OEIS has 3 
more terms than I give in UPINT 
<http://www.amazon.com/exec/obidos/tg/detail/-/0387208607/mathpuzzlecom/> 
(perhaps due to Flammenkamp?)."

If "a(n)=0 for all sufficiently large n" is false, then c would equal 2. 
But it's ?/sqrt(3), by Guy&Kelly.

Flammenkamp's record solution at n=52 still stands.

--Ed Pegg Jr

franktaw at netscape.net wrote:
> http://www.research.att.com/~njas/sequences/A000769.
>
> The Extensions contains the statement "It is known that a(n)=0 for all 
> sufficiently large n." However, I can find no support for this 
> statement from any of the referenced web sites, nor any other web 
> sites I could find - all refer to it as a conjecture. Since none of 
> the references is more recent than 1998, I doubt that any them prove 
> this, either.
>
> Is this really known? If so, what is the reference?
>
> Franklin T. Adams-Watters
>
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