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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.
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- W. W. Adams and D. Shanks, ``Strong primality tests that are not sufficient" Math. Comp. 39, pp. 255 - 300 (1982)
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