$2^m - 1$ as the Integer Formulation to Govern the Dynamics of Collatz-Type Sequences

It has been discovered that representing odd integers as modified binary expressions of the form $\sum \limits_{n > m}2^n + 2^m - 1$ for $m \geq 1$ helps in understanding the dynamics of Collatz-type sequences. Starting with the original Collatz sequence $3x + 1$, it is found that when the odd step is applied to an odd integer $\sum \limits_{n > m}2^n + 2^m - 1$, an even integer $3\left(\sum \limits_{n > m}2^{n}\right) + 2^{m + 1} + 2^m - 2$ is obtained, which is exactly once divisible by 2, unless the lowest index reduces to zero. This implies that the sequence alternates between odd and even steps $m$ times. This governs the dynamics of the Collatz-type sequences because the value of $m$ determines the number of times the integer can be divided by 2 in each even step. A shortcut method is developed based on this dynamics that states that the even integer after $m$ odd-even steps are completed is $\left(\left(\frac{3}{2}\right)^m \sum \limits_{n > m}2^n\right) + 3^m - 1$. A shortcut method of this magnitude has never been utilized anywhere. The shortcut method for the modified Collatz sequence $5x + 1$ is also presented.