Summary: 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 ("$generations$") 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.

05A15, 11B37, 11B39

meta-Fibonacci recursion, hofstadter sequence