So I don't really know where to go from here, or how to "guess a formula for an"
a0, a1,a2... is a sequence that a0 = a1 = 1 and, for n >= 1, an + 1 = n (an +an-1)
So I started off by doing the base case which is 1, and getting:
a1 + 1 = 1(a1 + a1-1) 1 + 1 = 1(1+1) 2 = 1(2) 2 = 2 Which checks out fine.
Then I assumed true for k, which was:
ak+1 = k(ak + ak -1)
Then I tried to prove for k+1, and got:
a(k+1)+1 = k+1(ak+1 + ak+1 -1)
In order to try to guess the formula, we calculate. We find that $a_2=2$, $a_3=6$, $a_4=24$, $a_5=120$, $a_6=720$.
This sequence looks familiar, up to $n=6$ we have $a_n=n!$.
Now prove by (strong) induction that $a_{n}=n!$ for all non-negative integers $n$.
Suppose that $a_{i}=i!$ for all $i\le k$. We show that $a_{k+1}=(k+1)!$. We have $$a_{k+1}=k(k!+(k-1)!)=k(k-1)!(k+1)=(k+1)!.$$ This completes the induction step.