**III. Padovan sequence**

Just like the *golden ratio* and *tribonacci constant*, powers of the *plastic constant* *P* can also be expressed in terms of sequences associated with it. *P* is a root of the equation,

or,

Define,

then powers of* P* are,

where *U* and *V* are the *Padovan* and *Perrin* sequences, respectively,

which start with index *n* = 0. Hence,

and so on. These sequences obey,

and their limiting ratio, of course, is *P*. While the Fibonacci sequence has a nice representation as a *square spiral*, the Padovan is a spiral of equilateral *triangles*,

The Perrin sequence also has a notable feature regarding *primality testing*. Let be the roots of,

then, starting with *n* = 0,

Indexed in this manner, if *n* is prime, then *n* divides . For example . However, there are *Perrin pseudoprimes*, composite numbers that pass this test, with the smallest being *n* = 521^2.

Lastly, like all the four limiting ratios of this family of recurrences, the plastic constant *P* can be expressed in terms of the *Dedekind eta function* as,

where,