Here are the slides used for the course Efficient experimental mathematics for combinatorics and number theory, given at the Vienna Summer School of Mathematics in Weissensee, Austria, September 23–27, 2019:

1. Lecture 1: Context, Motivation, Examples.
2. Problem session 1.
3. Lecture 2: Experimental Mathematics for Combinatorics.
4. Lecture 3: Inside the Experimental Math. Toolbox.
5. Problem session 2.

Here is some documentation on the gfun package.

And here is some documentation on a guessing package for Mathematica.

And below are some additional references:

1. Bailey and Borwein (2001): Experimental Mathematics: Recent Developments and Future Outlook.
2. Wilf (2007): Mathematics: an Experimental Science.
3. Borwein (2008): Implications of Experimental Mathematics for the Philosophy of Mathematics.
4. Bailey and Borwein (2011): Exploratory Experimentation and Computation.
5. Nemes, Petkovsek, Wilf and Zeilberger (1997): How to do Monthly problems with your computer.
6. Trefethen (2002): A Hundred-dollar, Hundred-digit Challenge.
7. Borwein (2005): The SIAM 100-Digit challenge- a study in high-accuracy numerical computing.
8. Bailey, Borwein, Kapoor and Weisstein (2006): Ten Problems in Experimental Mathematics.
9. Stenger (2017): Experimental Math for Math Monthly Problems.
10. Polya (1954): Induction in the Theory of Numbers (Chap. IV of Induction and Analogy in Mathematics).
11. Polya (1954): Euler's Most Extraordinary Law of the Numbers (Chap. VI of Induction and Analogy in Mathematics).
12. Andrews (1983): Euler's Pentagonal Number Theorem.
13. Preston (1991): The Mountains of Pi.