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Looking further down the sequence, there's a hint of periodicity which never seems to amount to much. Here are the first $999$ differences split into groups of $27$:

9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 7 8 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 8 7 9 9 6 9 9 6 9 8 7 9 
9 6 9 9 7 8 8 7 9 9 6 9 9 7 8 9 6 9 9 6 9 9 6 9 8 7 9 

UPD: Bullet51 discovered what seems to be a complete solution for the case $k = 3$. Understanding how and why it works may be the key to cracking the general case as well.

Looking further down the sequence, there's a hint of periodicity which never seems to amount to much. (Since this answer was becoming cluttered I've removed the huge table of differences, but one can probably find them in the edit history.)

UPD: Bullet51 discovered what seems to be a complete solution for the case $k = 3$. Understanding how and why it works may be the key to cracking the general case as well.

UPD: Following in Bullet51's steps I decided to try my hand at constructing some finite state machines for larger $k$ (see their answer below for the legend). This resulted in pictures I feel painfully obliged to share.

$k = 4$:

$k = 5$:

$k = 6$:

$k = 7$:

I've verified all of these FSMs for the first $10^7$ differences in each case. Hopefully someone can make sense of what's going on here.

UPD: Bullet51 discovered what seems to be a complete solution for the case $k = 3$. Understanding how and why it works may be the key to cracking the general case as well.

If we look at consecutive differences $\Delta_i = x_{i + 1} - x_i$, we get a sequence $9, 6, 9, 9, 7, 8, 8, 7, 9, 9, 6, 9, 9, 7, 8, 9, 6, \ldots$ It looks like it can be split into triples with sum $24$ (implying $x_{3i + 1} = 6 + 24i$ for whatever reason?).

If we look at consecutive differences $\Delta_i = x_{i + 1} - x_i$, we get a sequence $9, 6, 9, 9, 7, 8, 8, 7, 9, 9, 6, 9, 9, 7, 8, 9, 6, \ldots$ It looks like it can be split into triples with sum $24$ (implying $x_{3i + 1} = 6 + 24i$ for whatever reason?). Further update: a similar pattern seems to persist for any $k$: empirically $x_{ki + 1} = k(k + 1) / 2 + i(k^3 - k)$ for any integer $i \geq 0$.