Goddard

Binary strings with even number of 0's

found here https://people.cs.clemson.edu/~goddard/texts/theoryOfComputation/6a.pdf

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,

A011782 OEIS

$$\begin{cases} S \rightarrow 1S \ |\ 0A0S \ |\ \epsilon \\ A \rightarrow 1A \ |\ \epsilon \end{cases} \ \ \ \ \ \ \ \  S = 1 +X·S + Y·Lin(X)·Y·S $$

$$\begin{cases} S \rightarrow 1S \ |\ 0T \ |\ \epsilon \\ T \rightarrow 1T \ |\ 0S \end{cases} \ \ \ \ \ \ \ \begin{cases} S = 1 + X·S + Y·T \\ T = X·T + Y·S \end{cases} $$