The density of primes dividing a particular non-linear recurrence sequence
The density of primes dividing a particular non-linear recurrence sequence
Define the ECHO sequence $\{b_n\}$ recursively by $(b_0,b_1,b_2,b_3)=(1,1,2,1)$ and for $n\geq 4$, $$ b_n=\begin{cases} \dfrac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}} & \text{if }n\not\equiv 0\pmod 3,\\ \dfrac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}} &\text{if }n\e