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%I #22 Feb 15 2024 17:40:36
%S 1,1,3,8,62,535,4213
%N Number of prime triangle partitions of order n.
%C A triangle partition is prime if it does not contain a triangle partition of lower order.
%C The order of a triangle partition is the number of smaller triangles into which the initial triangle is divided. The sequence counts only topologically distinct partitions. - _M. F. Hasler_, Feb 14 2024
%H Ed Pegg Jr., <a href="http://www.mathpuzzle.com/triangle.html">Triangles</a>
%H Miroslav Vicher, <a href="http://www.vicher.cz/puzzle/triangles/triangles.htm">Triangle Partitions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangleDissection.html">Triangle Dissection</a>
%e From _M. F. Hasler_, Feb 14 2024: (Start)
%e a(2) = 1 because a triangle can be divided into two smaller triangles in only one way, up to topological equivalence, namely by a straight line going through one of the vertices and a point on the opposite side.
%e a(3) = 1 counts the dissection of a triangle ABC into three smaller ones by three segments AD, BD, CD, where D is a point inside ABC. There are three other topologically inequivalent partitions of order 3, each using two segments, as follows: {AE, AF}, {AE, EG} and {AE, BH}, where E and F are two distinct points on BC, G is a point on AB, and H is a point on AE. It is easy to see that these aren't prime since removing the smaller triangle that has side AC leaves a triangle partition of order 2. (End)
%Y Cf. A056814.
%K nonn,nice,hard,more
%O 2,3
%A _N. J. A. Sloane_, Sep 01 2000
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