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# Plastic constant

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The plastic constant or plastic number
 P
, also called the silver constant or silver number,[1] is the real root of the cubic equation
${\displaystyle x^{3}-x-1=0.\,}$
${\displaystyle P:={\sqrt[{3}]{{\frac {1}{2}}-{\sqrt {{\frac {1}{4}}-{\frac {1}{27}}}}}}+{\sqrt[{3}]{{\frac {1}{2}}+{\sqrt {{\frac {1}{4}}-{\frac {1}{27}}}}}}={\frac {{\sqrt[{3}]{9-{\sqrt {69}}}}+{\sqrt[{3}]{9+{\sqrt {69}}}}}{\sqrt[{3}]{18}}}.\,}$
 Where we used the cubic formula[2] to solve ${\displaystyle ax^{3}+bx^{2}+cx+d=0,\quad a\neq 0,\,}$ with ${\displaystyle a=1,\,}$ ${\displaystyle b=0,\,}$ ${\displaystyle c=-1,\,}$ ${\displaystyle d=-1,\,}$ giving the roots ${\displaystyle x=p+{\sqrt[{3}]{q-Q}}+{\sqrt[{3}]{q+Q}},\,}$ where ${\displaystyle p=-{\frac {b}{3a}}\quad \left(=0\right),\,}$ ${\displaystyle q=p^{3}-{\frac {pc+d}{2a}}\quad \left(={\frac {1}{2}}\right),\,}$ ${\displaystyle r={\frac {c}{3a}}\quad \left(=-{\frac {1}{3}}\right),\,}$ ${\displaystyle Q={\sqrt {q^{2}+(r-p^{2})^{3}}}\quad \left(={\sqrt {{\frac {1}{4}}-{\frac {1}{27}}}}\right).\,}$

## Decimal expansion of the plastic constant

${\displaystyle P=1.32471795724474602596090885447809734\ldots \,}$
A060006 Decimal expansion of real root of
 x 3  −  x  −  1
(sometimes called the silver constant, or the plastic constant).
 {1, 3, 2, 4, 7, 1, 7, 9, 5, 7, 2, 4, 4, 7, 4, 6, 0, 2, 5, 9, 6, 0, 9, 0, 8, 8, 5, 4, 4, 7, 8, 0, 9, 7, 3, 4, 0, 7, 3, 4, 4, 0, 4, 0, 5, 6, 9, 0, 1, 7, 3, 3, 3, 6, 4, 5, 3, 4, ...}

## Padovan sequence and Perrin sequence

The linear recurrence with constant coefficients of order
 3
${\displaystyle a(0)=a_{0},\,}$
${\displaystyle a(1)=a_{1},\,}$
${\displaystyle a(2)=a_{2},\,}$
${\displaystyle a(n)=a(n-2)+a(n-3),\quad n\geq 3,\,}$
with different choices of initial conditions gives the Padovan sequence (
 a0 = a1 = a2 = 1
) or the Perrin sequence (
 a0 = 3, a1 = 0, a2 = 2
).

The limit ratio of the recurrence gives the plastic constant

${\displaystyle \lim _{n\to \infty }{\frac {a(n+1)}{a(n)}}=P.\,}$

## Continued fraction and nested radicals expansions

The simple continued fraction expansion of the plastic constant is

${\displaystyle P=1+{\cfrac {1}{3+{\cfrac {1}{12+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}\,}$
A072117 Continued fraction expansion of smallest Pisot-Vijayaraghavan number (positive root of
 x 3 = x + 1
).
 {1, 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, 3, 1, 8, 2, 1, 1, 14, 1, 1, 2, 1, 1, 1, 3, 1, 10, 4, 40, 1, 1, 2, 4, 9, 1, 1, 3, 3, 3, 2, 1, 17, 7, 5, 1, 1, ...}

The plastic constant has the simplest nested cubic radicals expansion (the all one's sequence)

${\displaystyle P={\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}}}}}\,}$

since

${\displaystyle P^{3}=1+P.\,}$

If we consider the simplest nested square radicals expansion, we get the Golden ratio instead.

## Notes

1. Not to be confused with the silver ratio, i.e.
 1 + 2√  2
.
2. Weisstein, Eric W., Cubic Formula, from MathWorld—A Wolfram Web Resource.