Juggler sequences were first proposed by American mathematician and author Clifford A. Pickover. They take the form :

$${\displaystyle a_{k+1}={\begin{cases}\left\lfloor a_{k}^{\frac {1}{2}}\right\rfloor {\mbox{if }}a_{k}{\mbox{ is even}}\\\\\left\lfloor a_{k}^{\frac {3}{2}}\right\rfloor {\mbox{if }}a_{k}{\mbox{ is odd}}.\end{cases}}}$$

It's conjectured that all Juggler Sequences eventually reach 1. The sequences have been verified for initial terms up to 10⁶ - but the conjecture has yet to be proved or disproved.

Technical details Wolfram Mathworld

Also see : Collatz conjectureplugin-autotooltip__plain plugin-autotooltip_bigCollatz conjecture

Using the formula :

$$ {\displaystyle f(n)={\begin{cases}{\frac {n}{2}}\quad\quad\quad{\text{if }}n\equiv 0{\pmod {2}}\\[4px]3n+1\quad{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}} $$

The Collatz conjecture states that this process will eventually reach the number 1, regardless of which positive integer is chosen initially.

For example, using n = 12 generates the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.