  In this note we investigate the solutions of certain meta-Fibonacci
recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial
conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the
resulting integer sequence is closely related to the function counting binary
partitions of a certain type.
