What is $\sum\limits_{n=0}^{\infty} r^{an^2 + bn + c}$ ? or: is $0.0100100010000100001...$ transcendental?

Question

The idea is a more convenient form for $N = 0.01001000100001000001...$ in base $r$, hopefully to show whether it is transcendental.

Sorry for brevity.

Answer 1

Of course $N$ is transcendental. [I said it is, I didn't say I can prove it.] It is conjectured that all irrational algebraic numbers are normal in all bases. If this were not transcendental, it would be a spectacular counterexample to that conjecture.

Answer 2

This may come under the heading of "Siegel E-functions" or "Siegel G-functions", for which transcendence results are known.