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,