Is it possible to describe the Collatz function in one formula? This is related to Collatz sequence, which is that
$$C(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$
Is it possible to describe the Collatz function in one formula? (without modular conditions)

$$C(n)=\frac{n}{2}\cos^2\left(\frac{\pi n}{2}\right)+(3n+1) \sin^2\left(\frac{\pi n}{2}\right)$$
Continuous extension . M. Chamberland $(1996)$ observed that the entire function defined by 
$$f(x)=x+\frac{1}{4}-\frac{2x+1}{4} \cos (\pi x)$$
interpolates the $3n+1$ function: $$T(n)=\begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ (3n+1)/2 & \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $T(n)=f(n)$ for $n\in \mathbb{N}$. This allows him to try to apply methods of one-dimensional discrete dynamical systems to $3n+1$ iterations. 
Chamberland, M., A Continuous Extension of the 3x + 1 Problem to the Real Line, Dynamics of Continuous, Discrete and Impulsive Systems 2 A996), 495-509.
