Always great. A fine technique to compensate for those of us who lack mathematical sophistication: write some godawful brute force code to calculate the number

Always great. A fine technique to compensate for those of us who lack mathematical sophistication: write some godawful brute force code to calculate the number of items in some construct of interest, then search those numbers up to learn all sorts of useful new things about what you have. I learned about generalizations of the Fibonacci sequence and "half-Catalan numbers" by this method.