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We let $M(n)$ be the $3\times 3$ submatrix of $S$ which consists of the rows $3n-1,3n,3n+1$ of $S$, and let $T_0(n),T_1(n),T_2(n)$ be the $3 \times 3$ matrices \begin{align*} T_0(n):=\begin{pmatrix} 8n-4 & 8n-3 & 8n-2\\ 8n-2 & 8n-1 & 8n\\ 8n+1 & 8n+2 & 8n+3\\ \end{pmatrix},\;\; T_1(n):=\begin{pmatrix} 8n-4 & 8n-3 & 8n-3\\ 8n-2 & 8n-1 & 8n\\ 8n+1 & 8n+2 & 8n+3\\ \end{pmatrix}\,\;\end{align*} \begin{align*} T_2(n):=\begin{pmatrix} 8n-4 & 8n-3 & 8n-2\\ 8n-1 & 8n-1 & 8n\\ 8n+1 & 8n+2 & 8n+3\\ \end{pmatrix}\;\;. \end{align*}

By Cobham's theorem ("a sequence arising as (image under a coding of) a fixed point of a $k$-uniform morphism is $k$-automatic") the sequence $(t_n)$ is therefore $k$-automatic.
By the closure properties of the set of $k-$automatic sequences then also the sequence $(\Delta_n)$ with $$\Delta_n=(k^2-1)+t_{n+1}-t_n=x_{n+1}-x_n$$ is $k$-automatic. (That is, $\Delta_n$ can be computed from the base-$k$ digits of $n$ with a finite state machine of the type desribed above.)

We let $M(n)$ be the $3\times 3$ submatrix of $S$ which consists of the rows $3n-1,3n,3n+1$ of $S$, and let $T_0(n),T_1(n),T_2(n)$ be the $3 \times 3$ matrices \begin{align*} T_0(n):=\begin{pmatrix} 8n-4 & 8n-3 & 8n-2\\ 8n-2 & 8n-1 & 8n\\ 8n+1 & 8n+2 & 8n+3\\ \end{pmatrix},\;\; T_2(n):=\begin{pmatrix} 8n-4 & 8n-3 & 8n-3\\ 8n-2 & 8n-1 & 8n\\ 8n+1 & 8n+2 & 8n+3\\ \end{pmatrix}\,\;\end{align*} \begin{align*} T_1(n):=\begin{pmatrix} 8n-4 & 8n-3 & 8n-2\\ 8n-1 & 8n-1 & 8n\\ 8n+1 & 8n+2 & 8n+3\\ \end{pmatrix}\;\;. \end{align*}

By Cobham's theorem ("a sequence arising as (image under a coding of) a fixed point of a $k$-uniform morphism is $k$-automatic") the sequence $(t_n)$ is therefore $k$-automatic.
By the closure properties of the set of $k-$automatic sequences (under shifts, parallel generation and taking functions of output elements) then also the sequence $(\Delta_n)$ with $$\Delta_n=(k^2-1)+t_{n+1}-t_n=x_{n+1}-x_n$$ is $k$-automatic. (That is, $\Delta_n$ can be computed from the base-$k$ digits of $n$ with a finite state machine of the type desribed above.)

(1) the behaviour of $x_n$ is governed by the type sequence $t_n$, $x_n=8n-2$ if $t_n=0$, $x_n=8n-1$ if $t_n=1$, and $x_n=8n-3$ if $t_n=2$ (in the description above $t_n$ can be visualized as the column index of $x_n$)

(2) the sequence $t_n$ is the fixed point starting from $0$ of the $3-$uniform morphism $\phi$ of the free monoid $\{0,1,2\}^*$ given by $0\rightarrow 012,\;1\rightarrow 010,\;2\rightarrow 002$

(1) the behaviour of $x_n$ is governed by its type sequence $t_n$.

Then the sequence $t_n$ (with representatives $0,1,\ldots,k-1$ $\bmod k$) is the fixed point starting from $0$ of the $k-$uniform morphism $\phi$ of the free monoid $\{0,1,\ldots,k-1\}^*$ given by $p\rightarrow b(p)$.

Then the sequence $t_n$ is the fixed point starting from $\tfrac{k}{2}$ of the $k-$uniform morphism $\phi$ of the free monoid $\{0,1,\ldots,k-1\}^*$ given by $p\rightarrow b(p)$.

In view of Cobham's theorem ("a sequence arising as (image under a coding of) a fixed point of a $k-$uniform morphism is $k-$automatic") these findings explain the observations above.

(1) the behaviour of $(x_n)$ is governed by the type sequence $(t_n)$, $x_n=8n-2$ if $t_n=0$, $x_n=8n-1$ if $t_n=1$, and $x_n=8n-3$ if $t_n=2$ (in the description above $t_n$ can be visualized as the column index of $x_n$)

(2) the sequence $(t_n)$ is the fixed point starting from $0$ of the $3$-uniform morphism $\phi$ of the free monoid $\{0,1,2\}^*$ given by $0\rightarrow 012,\;1\rightarrow 010,\;2\rightarrow 002$

(1) the behaviour of $(x_n)$ is governed by its type sequence $(t_n)$.

Then the sequence $(t_n)$ (with representatives $0,1,\ldots,k-1$ $\bmod k$) is the fixed point starting from $0$ of the $k$-uniform morphism $\phi$ of the free monoid $\{0,1,\ldots,k-1\}^*$ given by $p\rightarrow b(p)$.

Then the sequence $(t_n)$ is the fixed point starting from $\tfrac{k}{2}$ of the $k$-uniform morphism $\phi$ of the free monoid $\{0,1,\ldots,k-1\}^*$ given by $p\rightarrow b(p)$.

By Cobham's theorem ("a sequence arising as (image under a coding of) a fixed point of a $k$-uniform morphism is $k$-automatic") the sequence $(t_n)$ is therefore $k$-automatic.
By the closure properties of the set of $k-$automatic sequences then also the sequence $(\Delta_n)$ with $$\Delta_n=(k^2-1)+t_{n+1}-t_n=x_{n+1}-x_n$$ is $k$-automatic. (That is, $\Delta_n$ can be computed from the base-$k$ digits of $n$ with a finite state machine of the type desribed above.)

(1) the behaviour of $x_n$ is governed by the type sequence $t_n$, $x_n=8n-2$ if $t_n=0$, $x_n=8n-1$ if $t_n=1$, and $x_n=8n-3$ if $t_n=2$ (in the description above $t_n$ can be visualized as the column of $x_n$)

(1) the behaviour of $x_n$ is governed by the type sequence $t_n$, $x_n=8n-2$ if $t_n=0$, $x_n=8n-1$ if $t_n=1$, and $x_n=8n-3$ if $t_n=2$ (in the description above $t_n$ can be visualized as the column index of $x_n$)