Is there a formula to construct a Collatz (3x + 1) sequence of arbitrary length that is strictly increasing? Obviously one can do this with a strictly decreasing sequence by just taking $2^n$ but I haven't seen a way to do it with a strictly increasing sequence.



Finding a strictly increasing Collatz sequence of arbitrary length
Start with a number equal to $-1$ modulo $2^n$. Then, after one step, the number is $\frac{3(-1)+1}{2}=-1$ modulo $\frac{2^n}{2}=2^{n-1}$, so inductively it will increase $n$ steps.


