Find the general term of the sequence defined by $x_0 = 3, x_1 = 4$ and $x_{n+1} = x_{n-1}^2- nx_n$ for all $n \in \mathbb N$

Question

Find the general term of the sequence defined by $$x_0 = 3, x_1 = 4\quad \&\quad x_{n+1} = x_{n-1}^2- nx_n \quad \forall n \in \mathbb N$$

I realized this is not a homogeneous recursion relationship. I don't know if it's the right expression, but it has variable coefficients, and it's not linear.

Is there another method to solve this without induction?

Answer 1

We prove by induction that $x_n=n+3$ for all $n\in\mathbb{N}$. Since $x_0=3=0+3$ and $x_1=4=3+1$, let $n\in\mathbb{N}$ such that $x_n=n+3$ and $x_{n+1}=n+4$, we have $$ x_{n+2}=x_n^2-(n+1)x_n=(n+3)^2-(n+1)(n+4)=n+5=(n+2)+3 $$ This ends the induction.