Why are the differences between consecutive squares equal to the sequence of odd numbers? I was playing around with the squares and saw an interesting pattern in their differences.
$0^2 = 0$
      + 1

$1^2 = 1$
      + 3

$2^2 = 4$
      + 5

$3^2 = 9$
      + 7

$4^2 = 16$
      + 9

$5^2 = 25$
      + 11

$6^2 = 36$
      etc.

(Also, in a very related question, which major Math Research Journal should I contact to publish my groundbreaking find in?)

In general, $a^2-b^2=(a-b)(a+b)$ So $(x+1)^2-x^2=(x+1-x)(2x+1)=2x+1$ So all odd numbers are generated.
