Consider the following recurrence relation
$z_{n} = c^2 + 2cz_{n-1}^2 + z_{n-1}^4 - (c+c^2)z_{n-1} - 2cz_{n-1}^3 - z_{n-1}^5$
where $z_{n}, c \in \mathbb{C}$.
I google a while but the formula for recurrence sequence, using the characteristic polynomial can be used for linear relation like Lucas Numbers.
I was thinking to moving on the continuos case so to solve the following non-linear differential equation over the complex field
$z'(x) = c^2 + 2cz^2(x) + z^4(x) - (c+c^2)z(x) - 2cz^3(x) - z^5(x)$
where $z$ is meromorphic function from complex to complex. Then come back to the discrete case.
If $z$ was a real-valued function this could be easly calculated by separating variables, but I do not know it I can apply the same procedure in the complex case.