Harmonic Triangle is a triangle wherein the first line
is the harmonic series, and each fraction in the triangle is the sum of
the term to the right of it, and below it.

1 
1/2 
1/3 
1/4 
1/5 
1/6 
... 

1/2 
1/6 
1/12 
1/20 
1/30 
... 


1/3 
1/12 
1/30 
1/60 
... 



1/4 
1/20 
1/60 
... 




1/5 
1/30 
... 





1/6 
... 





















It can be noted that each term on the harmonic triangle
is a fraction. Furthermore, each fraction is a reciprocal of a
natural number.
To build the harmonic triangle, simply fill the first
row and column with the harmonic series. Then fill in the values of
the remaining cells. Each of this cell has a value which is equal to
the number above it minus the number diagonally above it on the right.
For example, the value on the 2nd row and 2nd column is 1/6. This
number is equal to 1/21/3, number above it (1/2) minus the number
diagonally above it on the right (1/3).
A special property of the harmonic triangle is that each
fractions on the harmonic triangle is equal to the sum of all the fraction
below it and its right. For example, the second term on the first
row, 1/2, is equal to the sum of the series beginning with the number
below it.
1/2 = 1/6 + 1/12 + 1/20 + 1/30 .....
In Summary, the harmonic triangle has the following
properties:
1) The first line, as we know, is the harmonic series.
2) The fractions on the second line are one half the
reciprocls of triangular numbers. The fractions on the second line
sums up to one.
3) The fractions on the third line are 1/3 the
reciprocals of the pyramidal numbers.
4) Each fractions on the harmonic triangle is equal to
the sum of the fraction below it, and diagonally below it on the left.
5) Each fractions on the harmonic triangle is equal to
the difference of the fraction aboveit, and diagonally above it on the
right.
6) Each fractions on the harmonic triangle is equal to
the sum of all the fraction below it and its right.