%I #5 Mar 31 2012 21:08:36
%S 0,0,1,2,4,7,10,14,18,22
%N Maximal number of simple triangular regions that can be formed by drawing n line segments in the Euclidean plane.
%C 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 Inspection of Loy's web page shows that these are known to be optimal only for n up to about 7.
%C 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 David Coles, <a href="http://davcoles.tripod.com">Triangle Puzzle</a>.
%H Jim Loy, <a href="http://www.jimloy.com/puzz/cole.htm">Triangle Puzzle</a>.
%H Jim Loy, <a href="/A107427/a107427.gif">Illustration of a(6) = 7</a>
%e 7 lines can make at most 10 triangles, so a(7) = 10.
%Y Cf. A000124.
%K nonn,nice,more
%O 1,4
%A _Bill Blewett_, May 22 2005
%E Entry revised by _N. J. A. Sloane_, May 29 2005