Background
I was thinking about the Collatz conjecture, more specifically the mapping it implies. To show what I mean I define the following operators:
$$ \frac{\hat n}{2} = |2 \rangle \langle 1 | + |4 \rangle \langle 2| + | 6 \rangle \langle 3| + \dots $$
$$ (\hat{3n+1}) = |1\rangle \langle4 | + |3\rangle\langle 10| + |5\rangle\langle 16|+ \dots$$
These behave as functions in some sense (they are more accurately representing the mapping of a discrete function) as:
$$\langle 2 \lambda | \frac{\hat n}{2} = \langle \lambda| $$
Stating the Collatz conjecture as follows:
$$ \lim_{s \to \infty} (\langle4a| + \langle2a| +\langle a |) (\frac{\hat n}{2} + (\hat{3n+1}))^s = \langle4| + \langle2| +\langle 1 | $$
where $a$ is an arbitrary number. Now defining the $\hat a$ operator as:
$$ \hat a = |1\rangle \langle a | + | 2\rangle\langle 2a| + |3\rangle\langle 3a| + \dots $$
Using this definition:
$$ \lim_{s \to \infty} (\langle 4| + \langle 2| +\langle 1 |) \hat a (\frac{\hat n}{2} + (\hat{3n+1}))^s = \langle4| + \langle2| +\langle 1 | $$
Example
Focusing on ($s=1$):
$$ \hat a (\frac{\hat n}{2} + (\hat{3n+1})) $$ $$ = \frac{\hat n}{2} \hat a + \frac{n}{2}_\text{odd} \hat a + \hat a (\hat{3n+1}) $$ $$ = \frac{\hat n}{2} \hat a + \frac{n}{2}_\text{odd} \hat a' + (\hat{3n+1})\hat a \hat{(-3 a + 1)} $$
Where $\hat{(-3a +1)} = |3 a\rangle \langle 1| + |3a + 1 \rangle \langle 2| + |3a + 2\rangle \langle 3 | + \dots $ and $ \frac{n}{2}_\text{odd} = |1\rangle \langle \frac{1}{2}| + |3\rangle \langle\frac{3}{2}| + |5\rangle \langle \frac{5}{2}| +\dots $
and $\hat a ' = | \frac{1}{2}\rangle \langle \frac{a'}{2}| + |1\rangle \langle a'| + | \frac{3}{2}\rangle \langle \frac{3a'}{2}| +\dots $
Note: the above notation makes sense only if we double the dimension of our matrices. As $\langle m|\frac{n}{2}_\text{odd} |n\rangle = $ $2m$'th row and $2n$'th column.
Now focusing on the below operators:
$$ \hat{(-3 a + 1)} (\frac{\hat n}{2} + (\hat{3n+1})) $$ $$ = \hat{(-3 a + 1)} \frac{\hat n}{2} + \hat{(-3 a + 1)} (\hat{3n+1}) $$ $$ = \frac{\hat n}{2} \hat{(-3 a + 1)} \hat{(+\frac{3a+1}{2})} + (\hat{3n+1})\hat{(-3 a + 1)}(\hat{3n-2}) $$
Where $ \hat{(+\frac{3a+1}{2})} = |1 \rangle \langle 1 +\frac{3a+1}{2}| + |2 \rangle \langle 2| +\frac{3a+1}{2}| + |3 \rangle \langle 3|+\frac{3a+1}{2}| + \dots $ and $(\hat{3n-2}) = | 2 \rangle \langle 4| + | 3 \rangle \langle 7| + | 4 \rangle \langle 10 | + \dots $
Hence we can see a list of discrete functions emerging as we commute the operators as (for example consider $s=2$):
$$ \hat a (\frac{\hat n}{2} + (\hat{3n+1}))^2$$
$$\hat a (\frac{\hat n}{2} + (\hat{3n+1}) $$
$$ = (\frac{\hat n}{2} \hat a + \frac{n}{2}_\text{odd} \hat a'+ (\hat{3n+1})\hat a \hat{(-3 a + 1)})(\frac{\hat n}{2} + (\hat{3n+1})) $$ $$ = \frac{\hat n}{2} \hat a(\frac{\hat n}{2} + (\hat{3n+1})) + (\hat{3n+1})\hat a \hat{(-3 a + 1)}(\frac{\hat n}{2} + (\hat{3n+1})) $$
$$ = \frac{\hat n}{2} \hat a(\frac{\hat n}{2} + (\hat{3n+1})) + (\frac{n}{2}_\text{odd} \hat a')(\frac{\hat n}{2} + (\hat{3n+1}) + (\hat{3n+1})\hat a (\frac{\hat n}{2} \hat{(-3 a + 1)} \hat{(+\frac{3a+1}{2})} + (\hat{3n+1})\hat{(-3 a + 1)}(\hat{3n-2})) $$
Questions
If one proceeds with this strategy endlessly one will get a set of discrete functions. My question is there any formula or pattern for the functions arising from the $s$'th power of:
$$ \hat a (\frac{\hat n}{2} + (\hat{3n+1}))^s = ? $$
Also has anyone already devised this stratergy for studying the Collatz conjecture (reference request if yes)?