Question 136
The integer sequences $\{a_n\}$, $\{b_n\}$, $\{c_n\}$ are defined as follows: $a_1 = 3$, $b_1 = c_1 = 1$ and
$$
\begin{align*}
a_{n+1} &= 3b_n + c_n, \\
b_{n+1} &= a_n + b_n, \\
c_{n+1} &= b_n + c_n
\end{align*}
$$
for $n \ge 1$. Show that for any odd integer $N \ge 3$
i) there exists an index $m$ such that $a_m - 1, b_m$, and $c_m - 1$ are all divisible by $N$.
ii) at least one of $a_n - 1, b_n - 1$, and $c_n - 1$ is not divisible by $N$ for any index $n$.
(Otgonbayar Uuye)
$$
\begin{align*}
a_{n+1} &= 3b_n + c_n, \\
b_{n+1} &= a_n + b_n, \\
c_{n+1} &= b_n + c_n
\end{align*}
$$
for $n \ge 1$. Show that for any odd integer $N \ge 3$
i) there exists an index $m$ such that $a_m - 1, b_m$, and $c_m - 1$ are all divisible by $N$.
ii) at least one of $a_n - 1, b_n - 1$, and $c_n - 1$ is not divisible by $N$ for any index $n$.
(Otgonbayar Uuye)