Assuming you have a set of nodes, how do you determine how many connections are needed to connect every node to every other node in the set?
Example input and output:
In Out <=1 0 2 1 3 3 4 6 5 10 6 15
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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.