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.