an interesting new sequence!

[David Wilson]

The maximum number of triangles formable with n >= 1 lines is (n-1)(n-2)/2. 
This follows from the fact that n lines can be arranged so that any the form 
a triangle, which clearly achieves the maximum number possible.

The maximum number of triangular regions formable with n lines is another 
matter.  Rough bounds are (n+1)(n-2)/18 <= a(n) <= (n-1)(n-2)/2.  The lower 
bound is gotten from arranging 3n lines into a triangular grid.  The upper 
bound is the number of triangles which upper bounds the number of triangular 
regions.  a(n) is probably asymptotic to cn^2 with 1/18 <= c <= 1/2.

a(5) >= 5 because the pentacle arrangement includes 5 triangular regions. 
This means a(5) = 4 is incorrect and subsequenct elements are suspect.  I 
would admit elements to this sequence without an accompanying proof.

----- Original Message ----- 
From: "N. J. A. Sloane" <njas at research.att.com>
To: <seqfan at ext.jussieu.fr>
Sent: Friday, May 27, 2005 8:06 PM
Subject: an interesting new sequence!


>
> This just came in - at first I rejected it,
> thinking it was the same as A000124.  But it isn't!
>
> Can it really be new?
>
> NJAS
>
> %I A107427
> %S A107427 
> 0,0,1,2,4,7,10,14,18,22,27,32,38,44,50,54,60,72,76,84,92,110,114,122,
> %T A107427 130,156,160,210
> %N A107427 Maximal number of triangles that can be formed by n straight 
> lines in the Euclidian plane.
> %C A107427 A000124 is a related sequence, but that sequence refers to 
> regions whereas here we only consider triangles.
> %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>.
> %e A107427 7 lines can make at most 10 triangles, so a(7) = 10.
> %Y A107427 Cf. A000124.
> %K A107427 nonn,nice,more,new
> %O A107427 1,4
> %A A107427 Bill Blewett (billble(AT)comcast.net), May 22 2005
>