MHB Pythagorean Theorem: Unexpected Finding

[Angel11]

Hello, today i was playing around with the pythagorean theorem and found out something that i can't really explaing or atleast explain it with probably a false answer. So i was putting every possible combination with the max digit of 10. For example 1^2+1^2=\sqrt{2}, 1^2+2^2=\sqrt{5}... 10^2+1^2=\sqrt{101},10^2+2^2=\sqrt{104}. And then i found out that the square roots of the resolts are increasing by 3,5,7,9,11 and then i found out that those number are increasing by 2. so something like "1^2+2^2=\sqrt{(1^2+1^2)}+\sqrt{3} and then 1^2+3^2=\sqrt{(1^2+2^2)}+\sqrt{5}" so any idea how it works? my guess is WAY off i thought about it more and it is awful so i would appreciate anyones response.

 

[MarkFL]

If I understand correctly, you have observed for example:

$$1^2+0^2=1$$

$$1^2+1^2=2$$

$$1^2+2^2=5$$

$$1^2+3^2=10$$

And you've seen that the difference between successive equations are the sequence of odd natural numbers. Let's look at the difference between two successive equations in general:

$$1^2+n^2=n^2+1$$

$$1^2+(n+1)^2=n^2+2n+2$$

What do we get when we subtract the former from the latter?