A Number Triangle created by setting the Vertex equal to 0, filling one diagonal with 1s,
the other diagonal with multiples of an Integer , and filling in the remaining entries by summing the elements on
either side from one row above. Call the first column and the last column so that

(1) | |||

(2) |

then use the Recurrence Relation

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

Similarly, for we have

(10) |

(11) |

(12) |

So far, this has just been relatively boring Algebra. But the amazing part is that if is chosen as the
Integer, then and simplify to

(13) | |||

(14) |

which are consecutive Cubes and nonconsecutive Squares .

**References**

Clark, J. E. ``Clark's Triangle.'' *Math. Student* **26**, No. 2, p. 4, Nov. 1978.

© 1996-9

1999-05-26