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 $q_n=nϕ+O(1)$ or $q_n=n/ϕ+O(1)$, where $ϕ=(1+