Find all sequences that satisfy the recurrence relation
$$u_n\cdot (u_{n+1})^2-u_{n+1}-u_n+1=0, \text{with }u_0=1$$
My try
First, we find $u_1$, which follows $u_0=1$.
$u_0\cdot (u_{1})^2-u_{1}-u_0+1=0$
$(u_1)^2-u_1-1+1=0$
$(u_1)^2-u_1=0$
$u_1=0 \vee u_1=1$
Clearly, there are two possible sequences $a$ and $b$, which respectively begin with $1,0\ldots$ and $1,1,\ldots$.
Then, we find the $u_2$ for sequence $a$:
$u_1\cdot (u_{2})^2-u_{2}-u_1+1=0$
$-u_2-0+1=0$
$u_2=1$
Answer
I think the answer is:
$1,0,1,0,1,\ldots$ and $1,1,1,1,1,\ldots$
Question
Am I right?
You're basically there, but you seem to have forgotten that after a 1 you could have either a 0 or a 1. Basically it is all sequences of 1's and 0's where a one always follows a 0.