Question 194
Let $a_n, b_n$, $n = 1, 2, \ldots$ be two sequences of integers defined by $a_1 = 1$, $b_1 = 0$ and for $n \ge 1$,
$$
a_{n+1} = 7a_n + 12b_n + 6
$$
$$
b_{n+1} = 4a_n + 7b_n + 3.
$$
Prove that $a_n^2$ is the difference of two consecutive cubes.
$$
a_{n+1} = 7a_n + 12b_n + 6
$$
$$
b_{n+1} = 4a_n + 7b_n + 3.
$$
Prove that $a_n^2$ is the difference of two consecutive cubes.