Hydra function

The Hydra function is a Collatz-like function whose behavior is connected to the unsolved halting problems for the Cryptids Hydra and Antihydra. It is defined as: $$H(n)\equiv n+\lfloor \frac{1}{2}n\rfloor =\lfloor \frac{3}{2}n\rfloor =\{\begin{array}{ll}\frac{3n}{2}& \text{if }n\equiv 0(\mathrm{mod}2)\\ \frac{3n-1}{2}& \text{if }n\equiv 1(\mathrm{mod}2)\\ \end{array}$$ which can alternatively be written as $$\begin{array}{lll}H(2n)& =& 3n\\ H(2n+1)& =& 3n+1\\ \end{array}$$ It has some connections to Mahler's 3/2 problem.

Properties

Here, $s$ and $t$ are positive integers with $t$ odd. Let $0\le k\le s$ be an integer and $H^k$ is the $k$th iterate of $H$. $$\begin{array}{lll}{H}^k(2^{s}t)& =& 3^k{2}^{s-k}t\\ H^k(2^{s}t+1)& =& 3^k{2}^{s-k}t+1\\ \end{array}$$

Relationship to Hydra and Antihydra

Both machines effectively track the progress of two varibles; one of them changes depending on its value modulo 2 but roughly multiplies itself by $\frac{3}{2}$, and the other increases by 2 or decreases by 1 depending on the parity of the first variable.

In particular, Hydra halts if and only if the function Failed to parse (unknown function "\begin{cases}"): {\displaystyle f(x)=\begin{cases}\frac{3x+6}{2}&\text{if }a\equiv0\pmod{2}\\8}