X² + Y² : increasing by odd numbers

If you have the function N \times N \to N where f(x,y) = x^2 + y^2 and N stands for the set of natural numbers (starting at zero), then some interesting patterns emerge among the numbers.

Something that one can see with any sequence of numbers in this manner, i.e.,

3^2 + 1^2 = 10

3^2 + 2^2 = 13

3^2 + 3^2 = 18

The sum goes up by two more than the previous sum went up by. It starts at 3.

10 \to 13 = 3

13 \to 18 = 5

If you continued this sequence you could see growth of 7,9,11, and so on. I wonder why that is.

The same also happens with just one square number (1^2 = 1, 2^2 = 4, 3^2 = 9, and so on). I know this is nothing new, but I thought it was interesting to note.


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