let $\{s_n\}_{n=0,1,2...}$ be a strictly increasing sequence of positive integers, indexed by the finite ordinals. if $S$ denotes the space of all such sequences, let the accelerator function, $\alpha \in End(S)$, be defined by: $$ \alpha(s)_k = s_{s_k} $$ likewise define $\alpha^2$ by $$\alpha^2(s)_k = \alpha(\alpha(s))_k = s_{s_{s_k}} $$ and similarly for higher powers. using all of these, we may define $\alpha^{\omega}$, for the first limit ordinal $\omega$ by the rule: $$\alpha^{\omega}(s)_k = \alpha^k(s)_k $$ and then for the first successor ordinal of $\omega$: $$\alpha^{\omega+1}(s)_k = \alpha^{\omega}(s)_{\alpha^{\omega}(s)_k} $$and for the next limit ordinal: $$\alpha^{2\omega}(s)_k = \alpha^{\omega+k}(s)_{\alpha^{\omega+k}(s)_k} $$for the higher-order limit ordinal $\omega^2$ we define $$\alpha^{\omega^2}(s)_k = \alpha^{k\omega+k}(s)_{\alpha^{k\omega+k}(s)_k} $$and for the higher ordinal $\omega^{\omega}$ we write: $$\alpha^{\omega^{\omega}}(s)_k = \alpha^{k\omega^k +k\omega+k}(s)_{\alpha^{k\omega^k +k\omega+k}(s)_k} $$ all these sequences are subsequences of the original sequence s.
QUESTION if we begin with the sequence of squares, i.e. $s_k=k^2$ ($k=0,1,2,...$) so that we have $$\alpha^{\omega^{\omega}}(s)_0 =0 \\ \alpha^{\omega^{\omega}}(s)_1 =1 $$
what is the value of $$ \alpha^{\omega^{\omega}}(s)_2= ? $$