Prouhet-Thue-Morse constant

The Prouhet-Thue-Morse constant is the number $\tau$ whose binary expansion is the Prouhet-Thue-Morse sequence. That is,

$\tau=\sum_{i=0}^{\infty}\frac{t_{i}}{2^{i+1}}=0.412454033640\ldots$

where $t_{i}$ is the Prouhet-Thue-Morse sequence

Another expression, not in of $t_{i}$ , is

$\tau=\frac{1}{2}-\frac{1}{4}\prod_{n=0}^{\infty}(1-2^{-2^{i}})$

The number $\tau$ has been shown to be transcendental.