Another or the cornerstones of experimental math is the OEIS. The Online Encyclopedia of Integer Sequences. The page was started by Neil Sloane and Simon Pflouffe. It is wiser than the oracle at Delphi.
To show you how this works:
Supposing you had this question.
Is a(n) an integer for all n?
You cannot solve the recurrence, so you use it to generate some numbers. This is a typical procedure for experimental math.
1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188...
Looks like they will all be integers but we would like to be a little bit more certain of that. Copy the sequence and go here:
Paste the sequence into the top box and you get:
The Motzkin numbers! A well known combinatoric sequence.
So you look them up:
You see that the Motzkin numbers count the number of paths through a lattice. They are always integers. You cannot have 2.8 paths from some point to another on a lattice.
Then you look further and see the recurrence for the Motzkin numbers. If you take that recurrence and substitute n = n-1 into ours ( just a shift of indices ), it becomes that recurrence. We are done!
In mathematics, you don't understand things. You just get used to them.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.