an interesting new sequence!

[David Wilson]

I assume that the %S elements are proved optimal?

At any rate, my original mistake in interpreting this sequence led 
unwittingly to another puzzle.  What is the largest number of triangular 
regions that can be formed with n lines in the plane (lines go to infinity, 
nontriangular regions acceptable).


----- Original Message ----- 
From: "N. J. A. Sloane" <njas at research.att.com>
To: <seqfan at ext.jussieu.fr>; <njas at research.att.com>; "David Wilson" 
<davidwwilson at comcast.net>
Sent: Sunday, May 29, 2005 1:16 PM
Subject: Re: an interesting new sequence!


> David was commenting on the old version of that sequence.
>
> I revised it today.  Here is the current version.
>
> NJAS
>
> %I A107427
> %S A107427 0,0,1,2,4,7,10,14,18,22
> %N A107427 Maximal number of simple triangular regions that can be formed 
> by drawing n line segments in the Euclidean plane.
> %C A107427 Draw n line segments on a piece of paper in such a way that if 
> we make cuts along those lines, only triangular pieces are formed (apart 
> from the "outside" region).
> Sequence gives maximal number of triangles that can be obtained.
> %C A107427 Inspection of Loy's web page shows that these are known to be 
> optimal only for n up to about 7.
> %C A107427 Loy gives the following lower bounds for n = 1, 2, 3, ...: 0, 
> 0, 1, 2, 4, 7, 10, 14, 18, 22, 27, 32, 38, 44, 50, 54, 60, 72, 76, 84, 92, 
> 110, 114, 122, 130, 156, 160, 210
> %H A107427 David Coles, <a href="http://davcoles.tripod.com">Triangle 
> Puzzle</a>.
> %H A107427 Jim Loy, <a href="http://www.jimloy.com/puzz/cole.htm">Triangle 
> Puzzle</a>.
> %H A107427 Jim Loy, <a 
> href="http://www.research.att.com/~njas/sequences/a107427.gif">Illustration 
> of a(6) = 7</a>
> %e A107427 7 lines can make at most 10 triangles, so a(7) = 10.
> %Y A107427 Cf. A000124.
> %K A107427 nonn,nice,new,more
> %O A107427 1,4
> %A A107427 Bill Blewett (billble(AT)comcast.net), May 22 2005
> %E A107427 Entry revised by njas, May 29 2005
>