It's described explicitly in section 1 where they first reduce to a linear relationship and then recognize that a portion of the formula can be captured in a st

It's described explicitly in section 1 where they first reduce to a linear relationship and then recognize that a portion of the formula can be captured in a state variable, and rewrite as a recurrence relation. By persisting the state variable across subsequent computations they transform the quadratic formula for computing output into a linear formula computing output and next state from current state. It's kind of like memoization, but since it's a number it's constant space too.