Can every algebraic integer of degree $3$ be approximated by a quotient of linearly recurrent integer sequences of degree $3$? Given a zero $\alpha \in \mathbb{R}$ of an irreducible monic third degree polynomial $x^3 - a_2x^2 - a_1x - a_0$, are there always integer sequences $(p_n)_{n=1}^\infty$ and $(q_n)_{n=1}^\infty$ satisfying recursions
$$p_{n+1} = b_2p_n + b_1p_{n-1} + b_0p_{n-2} \\q_{n+1} = c_2q_n + c_1q_{n-1} + c_0q_{n-2},$$ with some integer coefficients, such that $\lim_{n\rightarrow \infty} \frac{p_n}{q_n} = \alpha$ ?
For quadratic irrationals, this is possible. Indeed, if $p(x) = x^2 - a_1x - a_0$ is the minimal polynomial of $\alpha$, and $p_n$ any sequence satisfying $p_{n+1} = a_1p_n + a_0p_{n-1}$, then the sequence $\frac{p_n}{p_{n-1}}$ converges to the zero with the bigger absolute value, and $\frac{a_0p_{n-1}}{p_n} $ converges to the smaller one, as long as $\alpha \neq \pm \sqrt{d}$. In that case the solutions of Pell's equation $x^2 - dy^2 = 1$ yield the desired sequences.
This trick also yields approximations for the largest and smallest real zero of any monic integer polynomial, but I couldn't find a similar trick to get at a possible "middle" zero.
 A: Assume that the polynomial $p(x)$ has three real roots $\alpha_1<\alpha_2<\alpha_3$.
Let $r\in\Bbb Q$ such that $\frac{\alpha_1+\alpha_2}2<r<\alpha_2$.
Then the largest root, in absolute value, of $q(x)=x^3p(r+1/x)$ is $\beta$ satisfying $\alpha_2=r+1/\beta$.
Write $r=u/v$ with $u,v\in\Bbb Z$ and $v>0$.
Then $v^3q(x)$ is an integer coefficients polynomial with leading term $v^3p(0)x^3$.
Consequently, $vp(0)\beta$ is the largest root, in absolute value, of a integer coefficients monic polynomial.
A stated in OP, there exists linear recurrence sequences $p_n,q_n$ of degree $3$ (satisfying the same linear recurrence) such that $p_n/q_n\to vp(0)\beta$.
Consequently,
$$\frac{p'_n}{q'_n}\xrightarrow{n\to\infty} r+1/\beta=\alpha_2$$
where
\begin{align}
p'_n&=up_n+v^2p(0)q_n\\
q'_n&=vp_n
\end{align}
which are linear recurrence sequences of degree 3.

As example, $1+2\cos(2\pi/7)$ is the largest root, in absolute value, of the polynomial $x^3-2x^2-x+1$, hence $p_n/q_n\to\cos(2\pi/7)$ where
\begin{align}
p_{n+3}&=2p_{n+2}+p_{n+1}-p_n&p_0&=b-a&p_1&=c-b&p_2&=c+b-a\\
q_{n+3}&=2q_{n+2}+q_{n+1}-q_n&q_0&=2a&q_1&=2b&p_3&=2c
\end{align}
for almost every choice of $a,b,c$.
