Cute problem I saw on quora:
If $$u_1 = 1,\qquad u_{n+1} = n+\sum_{k=1}^n u_k^2 $$ show that, for $n \ge 2$, $u_n$ is never a square.
\begin{align} n&=1:& u_2 &= 1+1 = 2\\ n&=2:& u_3 &= 2+1+4 = 7\\ n&=3:& u_4 &= 3+1+4+49 = 57\\ \end{align}
And, as usual, I have a generalization:
If $$a \ge 1,\quad b \ge 0,\quad u_1 = 1,\quad u_{n+1} = an+b+\sum_{k=1}^n u_k^2,$$ where $a, b \in \mathbb{Z}$, then for $n \ge 3$, $u_n$ is never a square.
Note: My solutions to these do not involve any explicit expressions for the $u_n$.