ISL 1982 Problem 52

We are given $2n$ natural numbers $$1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.$$ Find all $n$ for which these numbers can be arranged in a row such that for each $k \leq n$ , there are exactly $k$ numbers between the two numbers $k$ .