I have a diagram that shows a divisibility pattern between the numbers in the recursive sequence
$R_n = 2 R_{n-1} - 3R_{n - 2}$
with initial conditions $R_0 = 1$ and $R_1 = 0$, and the cumulative sum
$S_n = \sum_{k=0}^{n} R_{2k}$.
A cell with coordinates $x$ and $y$ is colored according to the value of the ratio $\log(\gcd(S_x, R_{2y}))/\log(|S_x|)$ if $S_x \neq 0$, and is colored blue if $S_x = 0$. Red corresponds to the value one and means that $S_x$ divides $R_{2y}$, while blue corresponds to zero and means that $S_x$ and $R_{2y}$ are relatively prime. Green indicates that $S_x$ and $R_{2y}$ have a greatest common divisor that is close to the square root of $|S_x|$. The leftmost column is for reference and is not colored according to this rule.
Greenish cells seem to occur regularly along certain lines. For example, the cell in $(x, y) = (16, 9)$ corresponds to $R_{18} = 12237$ and $S_{16} = 39937489$. Their greatest common divisor is $4079$, and $S_{16}$ factors as $4079 \cdot 9791$.
I have looked at lots of diagrams like this for various coefficients and initial conditions and many have a pattern like this one. Is there some theorem that explains the pattern? I'm asking because I wonder if a factorization method could be based on finding appropriate recurrence relations.