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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
Where we used the cubic formula[2] to solve

with

giving the roots

where

Decimal expansion of the plastic constant

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
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

Continued fraction and nested radicals expansions

The simple continued fraction expansion of the plastic constant is

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)

since

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

See also

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.

External links