Chains don't always rule

Proof sketch

All we need to do is determine the coefficient of the $x^{2016}$ term in $4\cos^3(x)$ , the rest isn't imporant because after taking the $2016^{th}$ derivative and plugging in $x=0$ all others terms will disappear.

We know $4\cos^3(x)=\cos(3x)+3\cos(x)$ .

Therefore the $x^{2016}$ term will be: $\frac{3^{2016}+3}{2016!}x^{2016}$

$\therefore \left. \frac{d^{2016}}{dx^{2016}} 4\cos^{3} x \right|_{x=0}=3^{2016}+3$

In base $3$ this is obviously $100...010$ consisting of only $\boxed{2}\, 1's$