This has probably been known for several thosand years, but not by me. (Well, ok, I'm not that old, but...) I found out several weeks ago when I was really bored. (and I do mean bored!)
(All variables are integers.)
Start with: n = x^2 - (x-1)^2
Expand: n = x^2 - x^2 + 2x - 1
Simplify: n = 2x - 1
Solve for x: x = { (n-1)/2 , (n+1)/2 }
Ok... well as a mathematical proof, that probably sucks goldfish, but it works for me! Now, I'm sure someone would have a use for that...
Also, for all multiples of four: x = { (n-4)/4 , (n+4)/4 }