Leibniz harmonic triangle

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The Leibniz harmonic triangle is a triangular arrangement of fractions in which the outermost diagonals consist of the reciprocals of the row numbers and each inner cell is the absolute value of the cell above minus the cell to the left. To put it algebraically, L(r, 1) = 1/n (where r is the number of the row, starting from 1, and c is the column number, never more than r) and L(r, c) = L(r - 1, c - 1) - L(r, c - 1).

The first eight rows are:

\begin{array}{cccccccccccccccccc} & & & & & & & & & 1 & & & & & & & &\\ & & & & & & & & \frac{1}{2} & & \frac{1}{2} & & & & & & &\\ & & & & & & & \frac{1}{3} & & \frac{1}{6} & & \frac{1}{3} & & & & & &\\ & & & & & & \frac{1}{4} & & \frac{1}{12} & & \frac{1}{12} & & \frac{1}{4} & & & & &\\ & & & & & \frac{1}{5} & & \frac{1}{20} & & \frac{1}{30} & & \frac{1}{20} & & \frac{1}{5} & & & &\\ & & & & \frac{1}{6} & & \frac{1}{30} & & \frac{1}{60} & & \frac{1}{60} & & \frac{1}{30} & & \frac{1}{6} & & &\\ & & & \frac{1}{7} & & \frac{1}{42} & & \frac{1}{105} & & \frac{1}{140} & & \frac{1}{105} & & \frac{1}{42} & & \frac{1}{7} & &\\ & & \frac{1}{8} & & \frac{1}{56} & & \frac{1}{168} & & \frac{1}{280} & & \frac{1}{280} & & \frac{1}{168} & & \frac{1}{56} & & \frac{1}{8} &\\ & & & & &\vdots & & & & \vdots & & & & \vdots& & & & \\ \end{array}

The denominators are listed in (sequence A003506 in OEIS), while the numerators, which are all 1s, are listed in A000012.

Just as Pascal's triangle can be computed by using binomial coefficients, so can Leibniz's: L(r, c) = \frac{1}{c {r \choose c}}. Furthermore, the entries of this triangle can be computed from Pascal's, "the terms in each row are the initial term divided by the corresponding Pascal triangle entries." (Wells, 1986)

This triangle can be used to obtain examples for the Erdős–Straus conjecture when n is divisible by 4.

[edit] References

  • D. Darling, "Leibniz' harmonic triangle" in The Universal Book of Mathematics: From Abracadabra To Zeno's paradoxes. Hoboken, New Jersey: Wiley (2004)
  • E. W. Weisstein, "Leibniz Harmonic Triangle." From MathWorld--A Wolfram Web Resource. [1]
  • D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 35.
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