Let $n\geq0$ be a positive integer and let $t_n$ denote the $n^{\rm th}$ element of the Thue-Morse sequence. Thus, $t_0=0,t_1=1,t_2=1,t_3=0, \ldots$.
Is there a formula for the integers $n$ such that $t_{n-1}=t_n=0$, i.e., 2 consecutive integers whose Thue-Morse value equals $0$?
EDIT
There is a sequence in the OEIS that describes the positions of two consecutive $0$s in the T-M sequence: A248056. Unfortunately it does not list any formula for this sequence.
For clarity, a function whose $n^{\rm th}$ value is the $n^{\rm th}$ term of A248056 would be sufficient.
For each nonnegative integer $n$ let $s_n$ denote the sum of the digits in the binary expansion of $n$. With this definition we have $$t_n = (s_n \bmod 2).$$
For nonnegative integers $n$ and $k$ let $n_k \in \{0,1\}$ denote the coefficient of $2^k$ in the binary expansion of $n$, and let $$i(n) = \inf\{j \in \mathbb{Z} \mid j\geq0 \land n_j=0\},$$ that is, $i(n)$ is the number of consecutive ones (counting starting from the least significant bit) occurring before the first zero in the binary sequence representing $n$ in the binary numeral system. e.g. $i(0)=0$, $i(1)=1$, $i(2)=0$, $i(3)=2$, etc.... With this definition it follows that for every nonnegative integer $n$ $$s_{n+1} = s_n - i(n) + 1.$$
If $n$ is a nonnegative integer then $$t_n = t_{n+1} = 0 \iff ( (s_n \bmod 2) = 0 \land (i(n) \bmod 2) = 1 ).$$ Therefore the integers we are interested in are those with an even number of ones in their binary expansion and such that there are an odd number of consecutive ones occurring before the first zero. Let $S$ denote the set of such integers.
For every nonnegative integer $n$ $n \in S$ if and only if there exist nonnegative integers $i$ and $j$ such that $j$ has an odd number of ones in its binary expansion and $$n = 2^{2i+2}j + 2^{2i+1} -1.$$
This is enough to easily check if a given $n$ is in $S$ and also enough to easily generate those $n$ which are in $S$. If you want a formula for a function that enumerates $S$ (a bijection from the nonnegative integers to $S$) that's a bit more work and the possible formulas depend on the functions you allow to be used in your formula.
It looks like you want an enumeration that's exactly equal to the sequence A248056, I believe this would be obtained by finding a strictly increasing enumeration of $S$ (well actually a bijection from positive integers to $S+1$ since you want $n$ such that $t_{n-1}=t_n=0$ as opposed to $t_n=t_{n+1}=0$). But once again the exact qualifying formulae depend on the operations allowed.