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?)
A nice thing to notice.

It's basically because

$(n + 1) ^ 2 - n ^ 2 = 2n + 1$
The difference between two consecutive cubes x^3 and (x+1)^3 will be 3x^2+3x+1.
It gets confusing after that... for consecutive hypercubes, it's 4x^3-6x^2+4x-1.