Iteration Sequence

A Sequence $\{a_j\}$ of Positive Integers is called an iteration sequence if there Exists a strictly increasing sequence $\{s_k\}$ of Positive Integers such that $a_1=s_1\geq 2$ and $a_j=s_{a_{j-1}}$ for $j=2$, 3, .... A Necessary and Sufficient condition for $\{a_j\}$ to be an iteration sequence is

$$a_j\geq 2a_{j-1}-a_{j-2}$$

for all $j\geq 3$.

References

Kimberling, C. ``Interspersions and Dispersions.'' Proc. Amer. Math. Soc. 117, 313-321, 1993.