The plastic number is well known to be the limiting ratio of the Padovan sequence (OEIS A000931), to wit,
$$ P_n=P_{n-2}+P_{n-3}\\ \lim_{n\to \infty} \frac{P_{n+1}}{P_n}=p $$
However, it is also the limiting ratio of the (unnamed) sequence (OEIS A003520)
$$ f_n=f_{n-1}+f_{n-5}\\ \lim_{n\to \infty} \frac{f_{n+1}}{f_n}=p $$
So the question is, how does it come to pass that two sequences have the same limiting ratio? Are there any other such examples?
What I have done:
I speculate that the this is because the plastic number is a morphic number, which is defined as follows:
$q$ is a morphic number $\Leftrightarrow q-1={{q}^{-n}}\wedge q+1={{q}^{m}},\,m,n\in \mathbb{N},\,m>1,\,q>0.$
In fact, there are only two morphic numbers, the golden ratio and the plastic number. That is
$$ \varphi+1=\varphi^2\quad \varphi-1=\varphi^{-1}\\ p+1=p^3\quad p-1=\varphi^{-4} $$
What distinguishes the plastic number from the golden ratio is that the morphic relations are different from each other, whereas for the golden the ratio they are the same.
Next I defined what I call the pseudomorphic numbers as those that satisfy only one of the morphic relations. These are denoted with the Greek letter chi, upper and lower case as follows:
$\chi$ is a pseudomorphic number $\Leftrightarrow \chi -1={{\chi }^{-n}},\,n\in \mathbb{N},\,n>0,\,\chi >0.$
$\text{X}$ is a pseudomorphic number $\Leftrightarrow \text{X} +1={{\text{X} }^{m}},\,m\in \mathbb{N},\,m>1,\,X >0.$
Next I demonstrated that all of the pseudomorphic numbers (which, of course, include the morphic numbers) are the limiting ratios of integer sequences, which were identified as follows:
For $\chi,\quad f_k=f_{k-1}+f_{k-1-n}$ (OEIS A000930 and related).
For $\text{X},\quad f_k=f_{k-m+1}+f_{k-m}$ (OEIS A103372 and related).
So, it can be seen here that the plastic number has two sequences, those for $m=3$ and $n=4$. You might wonder why the golden ratio, $m=2$ and $n=1$, does not have two sequences. However, you can readily determine that the two sequences are the same for these values of $m$ and $n$.
Of course, this is only an observation, not a proof. Can we prove that $p$ is the limiting ratio of two sequences and perhaps that this is a unique property?
FYI: I came to this question while examining the tiling properties of the pseudomorphic numbers, which I posted at the Tiling List. You can find additional information there.
The characteristic Polynomials of the two recursions are: $$ \eqalign{ & z^{\,3} - z - 1 \cr & z^{\,5} - z^{\,4} - 1 = \left( {z^{\,2} - z + 1} \right)\left( {z^{\,3} - z - 1} \right) \cr} $$
Both have only one real root, which is positive and greater than $1$, and which is in fact the plastic constant $\rho$. The asymptotic behaviour is therefore given by $$ P_{\,n} \approx f_{\,n} \approx \rho ^{\,n} \quad \left| {\;n \to \infty } \right. $$ and in particular the ratio of consecutive terms will tend to $\rho$ $$ \mathop {\lim }\limits_{n \to \infty } {{P_{\,n + 1} } \over {P_{\,n} }} = \mathop {\lim }\limits_{n \to \infty } {{f_{\,n + 1} } \over {f_{\,n} }} = \rho $$
Both have only one real root, which is positive and greater than $1$.
This will lead the asymptotic behaviour , and in particular will equal the ratio of consecutive terms.
-- additional note in reply to your comment --
A homogeneous linear recursion with constant coefficients has an associated "characteristic polynomial" (see the above link for a wider explanation).
If the roots $\rho_1, \cdots , \rho_m$ of the characteristic polynomial are, to make it simple, distinct, then the solution to the recursion is given by $$f_n = c_1 \rho_1^n +c_2 \rho_2^n + \cdots + c_m \rho_m^n$$ where the constants $c_k$ are determined by the initial conditions.
Then if $|\rho_1 | < |\rho_2 | < \cdots < |\rho_{m-1}| < 1 < |\rho_m|$, clearly $$f_n \; \to \; c_m \rho_m ^n$$ as $n \to \infty$ and $$\frac {f_{n+1}}{f_n} \; \to \rho_m$$ independently from the constants and thus from the initial conditions.
Therefore, all the solutions to linear homogeneous constant coefficients recursions which have the same predominant root $\rho_m$ will have the same asymptotic ratio, independently from the degree of the recursion and from the initial conditions.