A053740 Number of prime triangle partitions of order n.

1, 1, 3, 8, 62, 535, 4213

Offset: 2

Author

N. J. A. Sloane, Sep 01 2000

Keywords

• nonn
• nice
• hard
• more

Comments

A triangle partition is prime if it does not contain a triangle partition of lower order.

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

Examples

			From _M. F. Hasler_, Feb 14 2024: (Start)
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.
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)
		

Links

• Ed Pegg Jr., Triangles
• Miroslav Vicher, Triangle Partitions
• Eric Weisstein's World of Mathematics, Triangle Dissection

Crossrefs

Cf. A056814.