Formula for the number of connections needed to connect every node in a set?

If there are $n$ nodes, then this is called "$n$ choose $2$", and is equal to the number of $2$-element subsets of a set of $n$ elements. The Wikipedia article on binomial coefficients includes this and generalizations.

Since I started writing you discovered the correct formula. However, if you ever have a similar problem where you are trying to figure out a general form for the terms in a sequence from some initial values, a good tool is The On-Line Encyclopedia of Integer Sequences. In this case, entering 0,1,3,6,10,15 brings up a useful entry in which you can find the general form and references.

Here is what you want.$$\sum_{k=1}^{n-1}k=\frac{n(n-1)}{2}$$

Figured it out. The formula is:

x = n(n - 1) / 2