2ᵐ - 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 ∑ n > m2ⁿ + 2ᵐ - 1 for m ≥ 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 ∑ n > m2ⁿ + 2ᵐ - 1, an even integer 3(∑ n > m2ⁿ) + 2m + 1 + 2ᵐ - 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 ((3/2)ᵐ ∑ n > m2ⁿ) + 3ᵐ - 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.