A rational sinusoidal function

It is easy to get confused by the irrational nature of the sin function. One might think that you need irrational numbers to create a sinusoidal graph, but that's false. Consider $z_a = (a+i)^2/(a^2+1) \quad (a \in \mathrm{Rat})$. It lies on the unit circle in the complex plane and its coordinates are rational. Here is a rational sinusoidal function: $f(x) = \operatorname{Im}(z_a^x) \quad (x \in \mathrm{Int})$. This is what it may look like:

However, it has some irrational properties. According to WolframAlpha, $\lim_{x\to\infty}z_x^x = e^{2i}$. The function $f$ is probably injective for some values of $a$.

Ugliest sequence ever

Let $a_0 = 0, a_2 = 1$ and $a_n = a_{\left\lfloor{n/2}\right\rfloor} + a_{\left\lfloor{n/2}\right\rfloor+1}$.

Basic interpolation

Say we've got $n$ points through which we need to fit a polynomial. We'll solve the problem iteratively. First we'll find a polynomial that passes through one point, then we'll add a term to that polynomial to make it pass through another point, and then a third point will come into play -- etc.

Let the given points be $\begin{array}{c}n\\(x_i, y_i)\\i=1\end{array}$ and let $p_k$ denote the $k$th polynomial that we find. $p_n$ is our final answer. $p_k = \begin{cases}y_1 & \text{ if } k = 1\\ p_{k-1} + (y_k - p_{k-1}\bigg|_{x=x_k}) \prod_{i=1}^{k-1} \frac{x - x_i}{x_k - x_i} & \text{ if } 1 < k \leq n\end{cases}$.