A neat Mathematical connection

[Excubis]

Here is a dream I had couple weeks ago.
1) If you plot all perfect squares on a number line the number(elements) of numbers that separate them increase by 2, starting at 0.
2) Now if you take that number of elements(numbers) that separate perfect squares and divide that # by 2, and plot that quotient(answer), again on new number line you get all whole numbers.
example:
0,1..4....9......16........25..........36.............49..............64................81..................100....................121
0 2 4 6 8 10 12 14 16 18 20 (amount, of dots or numbers between each perfect square.
/2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 (divide by 2)
0 1 2 3 4 5 6 7 8 9 10 (answers)

I have personally done this to 4000 it works and is consistent. Just a neat connection to those interested in such things. Yes it is related to square of squares from Pythagoras.

[McCulloch]

[Replying to post 1 by Excubis]

So for the nth square number, the number of dots to the next square is

(n+1)^2 - n^2 - 1
The final minus one because you are counting neither endpoint.
Expanding:
=n^2+2n+1-n^2-1
=2n

[Excubis]

[Replying to post 2 by McCulloch]

Yup

[bluethread]

That is the nature of two dimensional geometry. Though my high school math teacher would scold me for doing this, in this case, visualizing the geometry shows that the algebra is not surprising. When on adds 1 to the number squared, one is adding one layer to two sides of the geometric square, with one unit overlapping on the corner. So, the natural result is that the difference is always 2 times the smaller number plus one, or two times the larger number minus one.