# 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.

## References

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

Title | Perrin sequence |
---|---|

Canonical name | PerrinSequence |

Date of creation | 2013-03-22 16:05:19 |

Last modified on | 2013-03-22 16:05:19 |

Owner | Mravinci (12996) |

Last modified by | Mravinci (12996) |

Numerical id | 5 |

Author | Mravinci (12996) |

Entry type | Definition |

Classification | msc 11B39 |

Defines | Perrin number |