Perrin sequence

Construct a recurrence relation with initial terms $a_0=3$, $a_1=0$, $a_2=2$ and $a_n=a_{n-3}+a_{n-2}$ for $n>2$. The first few terms of the sequence defined by this recurrence relation are: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119, 158, 209, 277, 367 (listed in A001608 of Sloane’s OEIS). This is the Perrin sequence, sometimes called the Ondrej Such sequence. Its generating function is

$$G(a(n);x)=\frac{3-x^2}{1-x^2-x^3}.$$

A number in the Perrin sequence is called a Perrin number.

It has been observed that if $n|a_n$, then $n$ is a prime number, at least among the first hundred thousand integers or so. However, the square of 521 passes this test.

The $n$th Perrin number asymptotically matches the $n$th power of the plastic constant.