AUSTRIAN-POLISH 1993 Problem 7

The sequence $(a_n)$ is defined by $a_0 = 0$ and $a_{n+1} = [\sqrt[3]{a_n +n}]^3$ for $n \ge 0$ . (a) Find $a_n$ in terms of $n$ . (b) Find all $n$ for which $a_n = n$ .