This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A053740 #23 Feb 16 2025 08:32:42
%S A053740 1,1,3,8,62,535,4213
%N A053740 Number of prime triangle partitions of order n.
%C A053740 A triangle partition is prime if it does not contain a triangle partition of lower order.
%C A053740 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 A053740 Ed Pegg Jr., <a href="http://www.mathpuzzle.com/triangle.html">Triangles</a>
%H A053740 Miroslav Vicher, <a href="http://www.vicher.cz/puzzle/triangles/triangles.htm">Triangle Partitions</a>
%H A053740 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangleDissection.html">Triangle Dissection</a>
%e A053740 From _M. F. Hasler_, Feb 14 2024: (Start)
%e A053740 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 A053740 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 A053740 Cf. A056814.
%K A053740 nonn,nice,hard,more
%O A053740 2,3
%A A053740 _N. J. A. Sloane_, Sep 01 2000