A Pattern in the Quadratic Formula!

I was going about finding the pattern of integers $a$ such that $x+a=x^2$ has at least one rational solution. Using the quadratic formula, the value boiled down to $\frac{1\pm\sqrt{1+4a}}{2}$. Thus, this occurs when $4a+1$ is the square of an integer.

I knew that odd squares can be expressed as $4a+1$ , which is not very hard to prove.

Consider $y^2-1$ , for some odd integer $y$ . Now, this can be factorized into $(y+1)(y-1)$ . Now, $2|y+1,2|y-1$ , since $y$ is odd. Thus, $4|y^2-1$ , or any odd $y^2$ can be expressed as $4a+1$ .

Now, I was making a table of odd integers and their squares to find any patterns. Only after $25$ , did I realise a pattern.

For $49$ , $a=12$ .

For $81$ , $a=20$ .

For $121$ , $a=30$ .

For $169$ , $a=42$ .

I realised a pattern, in which the difference between consecutive values of $a$ are in an arithmetic progression with difference of $2$ .

I tested my theory, and found it to comply, as for $225$ , $a=56$ , which I predicted.

Is their a proof for this theory? Did I stumble upon some intricate pattern?

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A Pattern in the Quadratic Formula!

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