Consider the following number:

$R=\frac{1}{9}\sum^\infty_{n=1} 10^{-\frac{n\left(n+1\right)}{2}}\left(10^n-1\right)\left(n\left(\operatorname{mod}10\right)\right)$

=0.122333444455555666666777777788888888999999999000000000011111111111222222222222333333333333344444444444444555555555555555666666666666666677777777777777777888888888888888888999999999999999999900000000000000000000...

which is formed by concatenating $n$ copies of $n\left(\operatorname{mod}10\right)$ after $0$.

The long sequences of repeating digits allow better and better rational approximations as the lengths of repeating digit blocks grow. Can this be a basis to prove this number transcendental?