Greedy queens and the golden ratio

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 $q_n=n\phi +O(1)$ or $q_n=n\mathrm{/}\phi +O(1)$, where $\phi =(1+\sqrt{5})\mathrm{/}2$ is the golden ratio.