There are two parts to this puzzle, but I am posting the other part tagged with ‘mathematics’ because it isn’t really a number sequence problem. Part I Find the next number in the sequence: 1,2,3,7,16,65,?
This is a sequence in OEIS, which makes it rather easy!
0, 1, 1, 2, 3, 7, 16, 65, 321, 4546, 107587
It's a bit like the Fibonacci series, but instead of $a(n) = a(n-1) + a(n-2)$, it's $a(n) = a(n-1) + a(n-2)^2$