Congruences modulo high powers of 2 for Sloane's box stacking function

We are given n boxes, labeled 1, 2, . . . , n. Box i weighs i grams and can support a total weight of i grams. The number of different ways to build a single stack of boxes in which no box will be squashed by the weight of the boxes above it is denoted by f(n). In a 2006 paper, the first author asked for “congruences for f(n) modulo high powers of 2”. In this note, we accomplish this task by proving that, for r ≥ 5 and all n ≥ 0, f(2n)− f(2r−1n) ≡ 0 (mod 2), and that this result is “best possible”. Some additional complementary congruence results are also given.