<s>[INST] Finding generating function and closed formula Find generating function and closed formula  $1,0,1,0,1,0,1,0,1,...$
Solution Attempt)
$$\begin{align}G(x) &=  1 + 0x + 1x^2 + 0x^3 + 1x^4 + 0x^5 + 1 x^6 + 0 x^7 + 1 x^8 + \dots\\
&= 1 + x^2  + x^4  + x^6 +   x^8 + \dots\end{align}$$
$$G(x) =  \sum_{k = 0}^{∞} x^{2k} = \sum_{k = 0}^{∞} (x^{2})^k = \frac{1}{1-x^2}$$
 [/INST] Following Ethan remark, we have
$$u_{2n}=1=|\sin((2n+1)\frac{\pi}{2})|$$
and
$$u_{2n+1}=0=|\sin((2n+2)\frac{\pi}{2})|$$
So, we can take
$$f(x)=|\sin((x+1)\frac{\pi}{2})|$$
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