Are any identities of the form $t(a+b)$ known, where $t(n)$ is the $n^{\rm th}$ element of the Thue-Morse sequence?

Question

Let $t(n)$ denote the $n^{\rm th}$ element of the Thue Morse sequence, i.e., $t(n)=1$ if the number of $1$s in the binary expansion of $n$ is odd, $0$ otherwise. Let also $t(0)=0$.

It is easy show that

• $t(2n)=t(n)$,
• $t(2n+1)=1-t(n)$.

Now let $a,b\in\mathbb{N}$ and greater than $0$. Can anything be proven about $t(a+b)$ for any specific (or general) values of $a$ and $b$? I have not found any identities of this form in the available literature.

Answer 1

If there are $c$ carries in the binary addition $a+b$, then $t(a+b) \equiv t(a) + t(b) + c \mod 2$.

EDIT: $c$ is also the $2$-adic valuation of ${a+b \choose a}$, by Kummer's theorem