Great question.  I've never tried this before, so here goes nothing:

If you don't understand this, or induction proofs in general, let me know.

This proof also outlines how you should come up with such summations.  We can do 1 to 5 by:

1: 1
2: 1 + 1
3: 2 + 1
4: 3 + 1
5: 3 + 2

Now we want to go from 5 to 8.  All we do is take our first 3, and add 5 to them.   We can do this since we know 5 can't be in any of them (since they are less than 5):

6: 5 + 1
7: 5 + 1 + 1
8: 5 + 2 + 1

Now we can do the same thing to get to 13:

9: 8 + 1
10: 8 + 1 + 1
11: 8 + 2 + 1
12: 8 + 3 + 1
13: 8 + 3 + 2

And by the proof above, this must work for all numbers.

I think that has to do with your screen resolution.  Try changing it and see if it helps.

The latex goes the page indeed.