derivation of plastic number

The plastic number may be defined to be the limit of the ratio of two successive members ( of the Padovan sequenceMathworldPlanetmath or the Perrin sequenceMathworldPlanetmath, both of which obey the recurrence relation

an=an-3+an-2. (1)

Supposing that such a limit

P:=limnan+1an (2)

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

anan-1an-1an-2=an-3an-2+1 (3)

and then let n.  It follows the limit equation


which is same as

P3=P+1. (4)

Thinking the graphs of the equations  y=x3  and  y=x+1, it is clear that the cubic equationMathworldPlanetmath

x3-x-1= 0 (5)

has only one real root (, which is P.

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

(u3+v3-1)+(3uv-1)(u+v)= 0.

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

{u2+v3= 1,uv=13,oru3v3=127

Thus u3 and v3 are the roots of the resolvent equation

z2-z+127= 0,



and accordingly


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



P=12(9+69)3+12(9-69)36. (6)

By (5), P is an algebraic integerMathworldPlanetmath of degree ( 3 (and a unit of the ring of integers of the number fieldMathworldPlanetmath (P)).  For computing an approximate value of P, see e.g. nth root by Newton’s method.

Title derivation of plastic number
Canonical name DerivationOfPlasticNumber
Date of creation 2013-03-22 19:09:41
Last modified on 2013-03-22 19:09:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Derivation
Classification msc 11B39
Related topic LimitRulesOfSequences
Related topic GoldenRatio
Related topic LimitRulesOfFunctions