Question 21
N4 (FRA) For any positive integer $x_0$, three sequences $\{x_n\}$, $\{y_n\}$, and $\{z_n\}$ are defined as follows:
(i) $y_0=4$ and $z_0=1$;
(ii) if $x_n$ is even for $n\ge0$, $x_{n+1}=\frac{x_n}{2}$, $y_{n+1}=2y_n$, and $z_{n+1}=z_n$;
(iii) if $x_n$ is odd for $n\ge0$, $x_{n+1}=x_n-\frac{y_n}{2}-z_n$, $y_{n+1}=y_n$, and $z_{n+1}=y_n+z_n$.
The integer $x_0$ is said to be good if $x_n=0$ for some $n\ge1$. Find the number of good integers less than or equal to $1994$.
(i) $y_0=4$ and $z_0=1$;
(ii) if $x_n$ is even for $n\ge0$, $x_{n+1}=\frac{x_n}{2}$, $y_{n+1}=2y_n$, and $z_{n+1}=z_n$;
(iii) if $x_n$ is odd for $n\ge0$, $x_{n+1}=x_n-\frac{y_n}{2}-z_n$, $y_{n+1}=y_n$, and $z_{n+1}=y_n+z_n$.
The integer $x_0$ is said to be good if $x_n=0$ for some $n\ge1$. Find the number of good integers less than or equal to $1994$.