Relation between series and equations There is following quotes from wiki on Plastic number:

The powers of the plastic number $A(n) = ρ^n$ satisfy the recurrence relation $A(n) = A(n − 2) + A(n − 3)$ for $n > 2$.

And 2nd is that 

Plastic number is the unique real solution of equation $x^3=x+1$

This last cubic equation and recurrence relation looks very similar if you assume that n represent 3rd power of x, correspondingly n-2 would represent x itself and n-3 would represent 1.  
But what is mathematical justification to associate the sequence and equation? 
I remember we did this trick in college to find formula for Nth Fibbonachi numbers (which is $A(n) = A(n − 1) + A(n − 2)$ and correspondingly $x^2=x+1$) but it was too long ago..
 A: The mathematical justification
is the standard association
of a linear recurrence 
with its associated
polynomial.
If the recurrence is
$\sum\limits_{k=0}^m c_k a_{n-k}
= 0
$
(with $c_0 \ne 0$),
if we assume that
$a_n = r^n$
with $r \ne 0$,
we get
$0
=\sum\limits_{k=0}^m c_k r^{n-k}
=r^n\sum\limits_{k=0}^m c_k r^{-k}
$,
so that
$1/r$
is a root of
$C(x)
=\sum\limits_{k=0}^m c_k x^{k}
$.
The initial conditions
for the $a_i$
then determine
which linear combination
of the roots
of $C(x)$
give the formula
for the $a_n$.
There are more complications
if $C(x)$
has repeated roots,
but this is why
the recurrence leads
to a polynomial.
Another nice result
of using the polynomial
is that the largest
(or maybe smallest)
root determines the
asymptotic growth
of the recurrence.
Look for generatingfunctionology
and download the book.
It's free
and extremely useful.
A: The formula 
$$\rho^n=A(n-2)+A(n-3)$$ 
is wrong, as a little computation shows. 
But the correct expression to get the sequence in A000931 is by taking
the three roots of $x^3-x-1=0$.
If $r,s,t$ are those (with $r$ the real one) then the correction is
$$A(n)=\frac{r^n}{2 r+3}+\frac{s^n}{2 s+3}+\frac{t^n}{2 t+3}.\qquad (1)$$
Obviously the entry in wikipedia gotta be corrected.
I still don't know how to deduce $(1)$ which is present in the aforementioned OEIS' entry, credited to Keith Schneider. 
A: This is uneccessarily complex and arcane. THE defining property of the plastic number is that it is a morphic number and satisfies the relations $p-1=p^{-4}$ and $p+1=p^3$. It the follows that $p^n=p^{n-1}+p^{n-5}$ and $p^n=p^{n+2}-p^{n-1}$. That's all there is to it. 
