A Triangle of numbers arranged in staggered rows such that

(1) |

(2) |

In addition, the ``Shallow Diagonals'' of Pascal's triangle sum to Fibonacci Numbers,

(3) |

where is a Generalized Hypergeometric Function.

Pascal's triangle contains the Figurate Numbers along its diagonals. It can be shown that

(4) |

(5) |

(6) |

In addition,

(7) |

It is also true that the first number after the 1 in each row divides all other numbers in that row Iff it is a
Prime. If is the number of Odd terms in the first rows of the Pascal triangle, then

(8) |

The Binomial Coefficient mod 2 can be computed using the XOR operation XOR , making Pascal's triangle mod 2 very easy to construct. Pascal's triangle is unexpectedly connected with the construction of regular Polygons and with the Sierpinski Sieve.

**References**

Conway, J. H. and Guy, R. K. ``Pascal's Triangle.'' In *The Book of Numbers.* New York: Springer-Verlag,
pp. 68-70, 1996.

Courant, R. and Robbins, H. *What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, p. 17, 1996.

Harborth, H. ``Number of Odd Binomial Coefficients.'' *Not. Amer. Math. Soc.* **23**, 4, 1976.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 31, 1983.

Pappas, T. ``Pascal's Triangle, the Fibonacci Sequence & Binomial Formula,'' ``Chinese Triangle,'' and ``Probability and Pascal's
Triangle.'' *The Joy of Mathematics.* San Carlos, CA: Wide World Publ./Tetra, pp. 40-41 88, and 184-186, 1989.

Sloane, N. J. A. Sequence
A007318/M0082
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.

Smith, D. E. *A Source Book in Mathematics.* New York: Dover, p. 86, 1984.

© 1996-9

1999-05-26