A diluted version of the problem of the existence of the Hofstadter sequence

We investigate the conditions on an integer sequence f(n)f(n), n is an element of N, with f(1) = 0, such that the sequence q(n), computed recursively via q(n) = q(n-q(n-1)) + f(n), with q(1) = 1, exists. We prove that f(n) is 'slow', that is, f(n+1) - f(n) is an element of {0,1}, n >= 1, is a sufficient but not necessary condition for the existence of sequence q. Sequences q defined in this way typically display non-trivial dynamics: in particular, they are generally aperiodic with no obvious patterns. We discuss and illustrate this behavior with some examples.