Leibniz harmonic triangle

Leibniz harmonic triangle

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:

egin{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 OEIS|id=A003506, while the numerators, which are all 1s, are listed in OEIS2C|id=A000012.

Whereas each entry in Pascal's triangle is the sum of the two entries in the above row, each entry in the Leibniz triangle is the sum of the two entries in the row "below" it. For example, in the 5th row, the entry (1/30) is the sum of the two (1/60)s in the 6th row.

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.

If one takes the denominators of the "n"th row and adds them, then the result will equal n 2^{n - 1}. For example, for the 3rd row, we have 3 + 6 + 3 = 12 = 3 × 22.

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. [http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html]
* D. Wells, "The Penguin Dictionary of Curious and Interesting Numbers." Penguin Books, NY, 1986, 35.


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