Patterns in square root whole numbers

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I am working with kids who discovered a pattern in squaring numbers and want to know why that is and is there a way of showing it as a formula and visually. The pattern is this: 1 squared is 1, 4 squared is 2, 9 squared is 3, 16 squared is 4, The pattern related to the number being squared and that it increased in a pattern of +3, +5, +7, and continues. Can you explain why?