NL-NWO 2009 Problem 2

Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$ , then adding $1$ again, then adding $2$ , and adding $2$ again, then adding $3$ , and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$ , then we have $a_0 = 0$ , and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$ . Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.