This is the start of the harmonic triangle:


1
1


1
2


1
2


1
3


1
6


1
3


1
4


1
12


1
12


1
4


1
5


1
20


1
30


1
20


1
5


1
6


1
30


1
60


1
60


1
30


1
6

...

Can you see what the rules for making the next line are?

Each fraction is equal to the sum of the two fractions below it.
Also, the nth row starts with the fraction 1/n.

Look at the triangle above and check that these rules really do work.

Can you work out the next two rows?

We can continue the first diagonal (1/1, 1/2, 1/3, 1/4, and so on) using the rule.
Can we be sure that the second diagonal (1/2, 1/6, 1/12, 1/20, and so on) can always be continued?

If not, try to find an example where it can't.

If we can, will it always contain unit fractions (that is, fractions where the number on top is 1)?
Can you prove it?