If I have a recurrence
$$R_n = aR_{n-1}+bR_{n-2}, R_1 = x, R_2 = y$$
and now I want to consider this new recurrence
$$Q_n = c(aQ_{n-1}+bQ_{n-2})$$
Is there any way to relate the two (perhaps by setting some specific initial values of $Q$)?
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I don't see an easy connection between the two sequences. If you choose for example $Q_2=R_2$ and $Q_1=R_1$ you get $Q_3=cR_3$ but $Q_4=c(acR_3+bQ_2)$ which is not $c^2 R_4$ because $Q_2=R_2 \neq c R_2$. – garondal Sep 01 '22 at 15:59
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Look at the roots of the corresponding characteristic polynomials. I don't think they are easily related. – Alexander Burstein Sep 01 '22 at 18:33