Prouhet-Thue-Morse constant

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

$$\tau =\sum _{i=0}^{\mathrm{\infty }}\frac{t_i}{2^{i+1}}=0.412454033640\mathrm{\dots }$$

where $t_i$ is the Prouhet-Thue-Morse sequence

Another expression, not in of $t_i$, is

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

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