Consider the Recurrence Relation

(1) |

(2) |

(3) |

(4) |

(5) |

For example, we have the sequences :

(6) |

(7) |

(8) |

(9) |

A sequence even more striking for remaining integral over many terms is the 3-Göbel sequence

(10) |

The Göbel sequences can be generalized to powers by

(11) |

**References**

Guy, R. K. ``The Strong Law of Small Numbers.'' *Amer. Math. Monthly* **95**, 697-712, 1988.

Guy, R. K. ``A Recursion of Göbel.'' §E15 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 214-215, 1994.

Sloane, N. J. A. Sequences
A003504/M0728
and A005166/M1551
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Zaiger, D. ``Solution: Day 5, Problem 3.'' http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.3.html.

© 1996-9

1999-05-25