Counting Odd Parity Substrings from Given Binary String

Problem

Give a Binary String of size $n$, return the number of Odd Parity substrings to be generated from a string.

Solution (Linear Time)

Assuming we are given a zero-indexed string $x$ Define a function

$$f(i)=\{\begin{array}{ll}0& \text{if }i=1\text{ and }x[0]=0\\ 1& \text{if }i=1\text{ and }x[0]=1\\ f(i-1)+g(i)& \text{if }1<i\le len(x)\end{array}$$ $$g(i)=\{\begin{array}{ll}0& \text{if }i=1\text{ and }x[0]=0\\ 1& \text{if }i=1\text{ and }x[0]=1\\ g(i-1)& \text{if }1<i\le len(x)\text{ and }x[i-1]=0\\ i-g(i-1)& \text{if }1<i\le len(x)\text{ and }x[i-1]=1\end{array}$$

Then, $f(len(x))$ will be the solution.