Question 174
Does there exist a non-constant sequence of natural numbers $a_1, a_2, \dots, a_n, \dots$ such that for every natural number $k \ge 2$ the equality
$$
a_k = \frac{2a_{k-1}a_{k+1}}{a_{k-1} + a_{k+1}}
$$
holds?
$$
a_k = \frac{2a_{k-1}a_{k+1}}{a_{k-1} + a_{k+1}}
$$
holds?