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:

$$\textstyle H(n)\equiv n+\left\lfloor {\frac {1}{2}}n\right\rfloor ={\Big \lfloor }{\frac {3}{2}}n{\Big \rfloor }={\begin{cases}{\frac {3n}{2}}&{\text{if }}n\equiv 0{\pmod {2}}\\{\frac {3n-1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}\\\end{cases}}$$

which can alternatively be written as

$${\begin{array}{l}H(2n)&=&3n\\H(2n+1)&=&3n+1\\\end{array}}$$

It has some connections to Mahler's 3/2 problem.

Properties

Here, $s$ and $t$ are positive integers with $t$ odd. Let $0\leq k\leq s$ be an integer and $H^{k}$ is the $k$th iterate of $H$.

$${\begin{array}{l}H^{k}(2^{s}t)&=&3^{k}2^{s-k}t\\H^{k}(2^{s}t+1)&=&3^{k}2^{s-k}t+1\\\end{array}}$$

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}