Question 156
For the real number sequence $\{a_n\}_{n=1}^\infty$, we are given that $a_1 = 1$, $a_2 = 3$, and for $n \ge 1$,
$$
a_{n+2} = a_{n+1} + \frac{3a_{n+1}^2 - 1}{a_{n+1} - a_n}
$$
Prove that all terms of the sequence $\{a_n\}$ are natural numbers and $a_{61} \gt 4^{61}$.
(Otgonbayar Uuye)
$$
a_{n+2} = a_{n+1} + \frac{3a_{n+1}^2 - 1}{a_{n+1} - a_n}
$$
Prove that all terms of the sequence $\{a_n\}$ are natural numbers and $a_{61} \gt 4^{61}$.
(Otgonbayar Uuye)