derivation of plastic number

The plastic number may be defined to be the limit of the ratio of two successive members (http://planetmath.org/Sequence) of the Padovan sequence or the Perrin sequence, both of which obey the recurrence relation

$a_n=a_{n-3}+a_{n-2}.$(1)

Supposing that such a limit

$P:=\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{a_{n+1}}{a_n}}$(2)

exists (and is $\ne 0$), we first write (1) as

${\displaystyle \frac{a_n}{a_{n-1}}}\cdot {\displaystyle \frac{a_{n-1}}{a_{n-2}}}={\displaystyle \frac{a_{n-3}}{a_{n-2}}}+1$(3)

and then let $n\to \mathrm{\infty }$.  It follows the limit equation

$$P\cdot P=\frac{1}{P}+1,$$

which is same as

$P^3=P+1.$(4)

Thinking the graphs of the equations  $y=x^3$  and  $y=x+1$, it is clear that the cubic equation

$x^3-x-1=\mathrm{\hspace{0.33em}0}$(5)

has only one real root (http://planetmath.org/Equation), which is $P$.

For solving the plastic number from the cubic, substitute by Cardano (http://planetmath.org/CardanosFormulae) into (5) the sum $x:=u+v$  of two auxiliary unknowns, when the equation may be written

$$(u^3+v^3-1)+(3uv-1)(u+v)=\mathrm{\hspace{0.33em}0}.$$

Then, as in the example of solving a cubic equation, $u$ and $v$ are determined such that the first two parentheses vanish:

$\{\begin{array}{cc}{u}^2+v^3=\mathrm{\hspace{0.33em}1},\hfill & \\ uv=\frac{1}{3},\text{or}\mathit{\hspace{1em}}{u}^3{v}^3=\frac{1}{27}\hfill & \end{array}$

Thus $u^3$ and $v^3$ are the roots of the resolvent equation

$$z^2-z+\frac{1}{27}=\mathrm{\hspace{0.33em}0},$$

i.e.

$$z=\frac{9\pm \sqrt{69}}{18}$$

and accordingly

$$u=\sqrt[3]{\frac{9+\sqrt{69}}{18}},v=\sqrt[3]{\frac{9-\sqrt{69}}{18}}.$$

Fixing that these the real values of the cube roots, we obtain the value of the plastic number in the form

$$x=\sqrt[3]{\frac{9+\sqrt{69}}{18}}+\sqrt[3]{\frac{9-\sqrt{69}}{18}},$$

or

$P={\displaystyle \frac{\sqrt[3]{12(9+\sqrt{69})}+\sqrt[3]{12(9-\sqrt{69})}}{6}}.$(6)

By (5), $P$ is an algebraic integer of degree (http://planetmath.org/DegreeOfAnAlgebraicNumber) 3 (and a unit of the ring of integers of the number field $\mathbb{Q}(P)$).  For computing an approximate value of $P$, see e.g. nth root by Newton’s method.