\begin{abstract}
We completely solve the meta-Fibonacci recursion 
$$V(n) = V(n - V(n - 1)) + V(n - V(n - 4)),$$
a variant of Hofstadter's meta-Fibonacci
$Q$-sequence. For the initial conditions $V(1) = V(2) = V(3) = V(4) =
1$ we prove that the sequence $V(n)$ is monotone, with successive terms
increasing by 0 or 1, so the sequence hits every positive integer. We
demonstrate certain special structural properties and fascinating
periodicities of the associated frequency sequence (the number of times
$V(n)$ hits each positive integer) that make possible an iterative
computation of $V(n)$ for any value of $n$.  Further, we derive a
natural partition of the $V$-sequence into blocks of consecutive terms
(``\emph{generations}'') with the property that terms in one block
determine the terms in the next. We conclude by examining all the other
sets of four initial conditions for which this meta-Fibonacci recursion
has a solution; we prove that in each case the resulting sequence is
essentially the same as the one with initial conditions all ones.
\end{abstract}