No 3 in line problem

[Richard Guy]

That there are only finitely many solutions
with  2n  points, no 3 in line, is just a
conjecture, supported by the heuristic
reasoning in the Guy & Kelly paper.   R.

On Tue, 5 Dec 2006, Ed Pegg Jr wrote:

> 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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