The Thue–Morse sequence$^{[1]}$$\!^{[2]}$ $t_n$ is an infinite binary sequence constructed by starting with $t_0=0$ and successively appending the binary complement of the sequence obtained so far: $$\begin{array}l 0\\ 0&\color{red}1\\ 0&1&\color{red}1&\color{red}0\\ 0&1&1&0&\color{red}1&\color{red}0&\color{red}0&\color{red}1\\ 0&1&1&0&1&0&0&1&\color{red}1&\color{red}0&\color{red}0&\color{red}1&\color{red}0&\color{red}1&\color{red}1&\color{red}0\\ \hline 0&1&1&0&1&0&0&1&1&0&0&1&0&1&1&0&1&0&0&1&0&1&1&\dots\\ t_0&t_1&t_2&t_3&t_4&\dots\!\!\! \end{array}$$
It has many interesting properties: it is aperiodic, cube-free, shows the parity of the number of $1$'s in the binary representation of a natural number, has connections to the Fabius function, the hypergeometric function, etc.
There is a nice formula for this sequence that uses only elementary functions, binomial coefficients and finite summation: $$t_n=\frac43\,\sin^2\left(\frac\pi3\left(n-\sum_{k=1}^n(-1)^{\binom n k}\right)\right)=\operatorname{mod}\left(2n+\sum_{k=1}^n(-1)^{\binom n k},\,3\right).$$ Unfortunately, I could not find a proof of this formula anywhere and could not construct it myself. So, I'm asking for your help with this.