Everything about The Leibniz Harmonic Triangle totally explained
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:
. 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.
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