This induces the Catalan number, the answer is
see
Catalan number - Wikipedia, the free encyclopedia
There are 2n points marked on a circle. We want to divide them into pairs and connect the points in each pair by a segment (chord) such that these segments do not intersect.
Let be the number of ways to do this.
Write a recurrence relation for .
I worked out up to n=5:
I had worked out to be but this failed when I worked out n=5.
This induces the Catalan number, the answer is
see
Catalan number - Wikipedia, the free encyclopedia
For the recurrence relation, you may choose the point numbered 1.
1 must be connected with some point. you may assume it is 2k(why?). Thus the 2n points are divided into two parts, one is 1,2,...2k, the another is 2n,2n-1,...2k+1. For any x in the first part,y in the second part, x,y cannot be connected since 1,2k is connected. So in this situation, there will be answers. So
Of course, if you want to deduce the previous formula from this recurrence relation, it is extremely tough. You may challenge it!
Hello, DaRush19!
I've verified your values up to .There are points marked on a circle.
We want to divide them into pairs and connect the points in each pair
by a segment (chord) such that these segments do not intersect.
Let be the number of ways to do this.
Write a recurrence relation for .
I worked out up to
. .
. . Is it possible that: ?
If so, the recurrence seems to be: .