Continued fractions with partial quotients bounded in average

We ask, for which n does there exists a k, 1 <= k <= n and (k, n) = 1, so that k/n has a continued fraction whose partial quotients are bounded in average by a constant B? This question is intimately connected with several other well-known problems, and we provide a lower bound in the case of B = 2. The proof, which is completely elementary, involves a simple "shifting" argument, the Catalan numbers, and the solution to a linear recurrence.