DE-BUNDESWETTBEWERB 2010 Problem 6

The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor$ for $n \ge 1$ . Find all numbers that appear more than twice at this sequence.