The Catalan numbers are a sequence of numbers, much like the Fibonacci numbers, which are given by the equation

Like the Fibonacci numbers, they too pop up all over the place, for example, the Catalan numbers correspond to the number of ways a regular n-gon can be divided into n-2 triangles.
Above is a visualization of the Catalan numbers.

A polyhedron constructed by folding along the edges of a Spidron - a figure in the plane made up of two alternating sequences of isosceles triangles.

A visual proof that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1