I have the following recurrence
$$T(n) = \frac{1 - T(n - 1)}{T(n - 2)}$$
and $T(0) = a, T(1) = b$. I re-wrote it as such
$$T(n)T(n - 2) = 1 - T(n - 1)$$
I am now trying to re-write this in linear form by making some sort of substitution. What could I do to make this recurrence linear?
This isn’t exactly an answer to your question, but I think it’s relevant. I doubt there will be a nice way to express $T$ linearly, because it turns out that $T$ is actually periodic. This becomes clear after calculating the first couple of terms: $$\color{green}{T(2)=\frac{1-b}{a}}$$ $$\color{green}{T(3)=\frac{a+b-1}{ab}}$$ $$T(4)=\frac{1-a}{b}$$ $$T(5)=\frac{a+b-1}{ab}$$ $$\color{green}{T(6)=\frac{1-b}{a}}$$ $$\color{green}{T(7)=\frac{a+b-1}{ab}}$$