It always struck me as obvious; with a square of n*n, just add n + n (for the two sides) + 1 (for the corner) and you get the next square. 2n+1 progresses 1, 3, 5 etc., plain as day.
What got me thinking about it (at perhaps 11 or 13 or so) was trying to turn it into a way of calculating the square root.
Wait, really? You never worked out in school (or on your own before it ever came up in school) that
and likewise that and ? (of course, it also comes up geometrically if you start looking at the number of little squares in a big square, and count the next row)