On the behavior of a family of meta-fibonacci sequences

A family of meta-Fibonacci sequences is defined by the k-term recursion $$ T_{a,k}(n) :=\sum_{i=0}^{k-1}T_{a,k}({n-i-a-T_{a,k}(n-i-1)}), \quad n>a+k,\,k\ge2, $$ with initial conditions $T_{a,k}(n)=1$ for $1\le n \le a+k$. Some partial results are obtained for $a\ge 0$ and $k>1$. The case a=0 and k odd is analyzed in detail, giving a complete characterization of its structure and behavior, marking the first time that such a parametric family of meta-Fibonacci sequences has been solved. This behavior is considerably more complex than that of the more familiar Conolly sequence (a=0, k=2). Various properties are derived: for example, a certain difference of summands turns out to consist of palindromic subsequences, and the mean values of the functions on these subsequences are computed. Conjectures are made concerning the still more complex behavior of a=0 and even k > 2.