Hi. I'm looking to find integer pairs $(a,c)$ such that $4a^2 + 8c^2 - 4c +1$ is a perfect square.

The sum is odd so I set the sum equal to $(2n+1)^2$ to cancel out the 1s and I end up with $\boxed{a^2 + 2c^2 - c = n^2 + n}$.

I haven't found a valid technique to break this down yet but I think it could be a pell's equation in disguise. I'm wondering if anyone can give me some insight on this problem. Thanks.