Dekaheptoid

Dekaheptoid (from Ancient Greek word meaning "seventeen") is an informal class of Turing machines that are, in some sense, similar to Skelet 17. A typical Turing machine in this class has the following behavior:

• it works with the sequence of numbers encoded as $10^{k}$ (here referred to as digits).
• it spends most of it's time performing an operation that could be described as incrementing/decrementing Gray Code of the sequence
• it has a roughly cubic growth rate

The specifics can differ, but some of common tape transformations and conditions are:

• Increment: the value at the last index is decremented and some other value is incremented. Typically this does not expand the tape.
• Halve: the last digit (typically having a value 1 or less) is deleted.
• Zero: expand the tape to the left by adding some number of leading zeros. The leftmost digit could be incremented by some small value. Typically this occurs when all digits (possibly, except for the least significant one) are even.
• Overflow: incrementing the leftmost digit creates new leading zeros.
• Smudge: if an "internal" (i.e. neither the first nor the last) digit is zero, increment it and decrement some digit next to it. In some cases leads to halting.

5-state dekaheptoids

Two notable machines are Skelet 17 and [1] which is equivalent to Skelet 17 starting from the state D and halts after 533 steps (29 rule applications).

6-state dekaheptoids

Known dekaheptoids fall into one of 12 equivalence classes.

2-state, 5-symbol dekaheptoids

The 2LA1RA4LB2LA2RA machine encodes digits as $4^{k}$ separated by $12$. For example, [x,y,z] is represented as $1<B4^{x}124^{y}124^{z}$. Note this machine is mirrored (z is the most significant digit while x is the least significant).

The analysis by @dyuan01 is available here: [2]