Question 104
Let $a_1 = 1$, $b_1 = 0$, $c_1 = 1$ and define for all $n \ge 1$
$$
a_{n+1} = a_n + 2b_n, \quad b_{n+1} = b_n + 2c_n, \quad c_{n+1} = a_n + c_n.
$$
Find integers $A$, $B$, $C$ such that $a_{n+3} = Aa_{n+2} + Ba_{n+1} + Ca_n$ for all $n \ge 1$.
$$
a_{n+1} = a_n + 2b_n, \quad b_{n+1} = b_n + 2c_n, \quad c_{n+1} = a_n + c_n.
$$
Find integers $A$, $B$, $C$ such that $a_{n+3} = Aa_{n+2} + Ba_{n+1} + Ca_n$ for all $n \ge 1$.