Greedy queens and the golden ratio

Source abstract

Place a queen in each successive column of an infinite $\mathbb{N}\times\mathbb{N}$ chessboard, always choosing the lowest row such that no two queens may attack one another. We prove that the row $q_n$ occupied by the queen in the $n$th column satisfies $q_n=nφ+O(1)$ or $q_n=n/φ+O(1)$, where $φ=(1+\sqrt5)/2$ is the golden ratio.