Here are my initial impressions when opening the apps for the first time. Not sure what the three icons are for at the bottom. Heart is not a good choice for "currently treanding', and neither is the down arrow in the box for "latest submission". I thought these were for favorite and download respectively.
I also didn't like the checkmark icon to go back when viewing an article.
Otherwise, I like the little animations, and everything runs smoothly. Good job.
When i receive such a request i use the object api http://develop.github.com/p/object.html to get a list of files in the repo, and then fetch *.textile and .png files which are new or have a new hash. I don't use their libraries, i just consume the rest interface directly.
I store the files in mongodb together with some metadata, and for processing i use a nice scala web framework called liftweb. For textile parsing i use http://www.eclipse.org/mylyn/downloads/ And for code formatting i use google-code-prettify
picture it like finding the area of a trinagle that is half of a square (long side goes from diagonally opposite corners)
formula for area of a square is n^2, so we need half that.
What about +n part of it? It is because we are dealing in discrete chunks, not smoothly continuous.
Alternately, imagine a series of children's blocks
now we have a 'triangle' with left side 3 high and bottom 3 wide. IF we make an identical triangle and rotate it round so that they fit together to make a rectangle, it looks like this
1 + 3
2 + 2
3 + 1
This makes a rectangle 3 x 4, and we should be able to figure out that this gives us an area of 12. So the original triangle ( 1,2,3 ) gives us half that and thus the sum from 1 to 3 is 6.
So the pattern for summing the numbers 1 through N is to take a square of length N, add 1 to one of its sides, so it has sides of length N and N+1, and then find the area (which is always N^2 + N) and then halve that.