I am trying to find integers $N$ that satisfy the following relationship:
$$N^2 - N = 2a^2 + 2a$$
for some integer $a$. I have computationally verified that such numbers do exist. But finding large values of such $N$ is computationally expensive. It feels like this is a simple undergrad discrete math problem, but I am having an unusally hard time finding solutions to this.
Assuming you want $n$ and $a$ positive:
$$ (2n-1)^2 - 2 (2a+1)^2 = -1 $$ The values of $2n-1$ are in the sequence $1, 7, 41, 239, 1393, 8119,$ with rule $s_{j+2} = 6 s_{j+1} - s_j.$ So the values of $n$ are $1, 4, 21, 120, 697$ with rule $$n_{j+2} = 6 n_{j+1} - n_j -2.$$
The values for $a$ are $0,2,14, 84, 492, 2870,$ and obey $$a_{j+2} = 6 a_{j+1} - a_j +2.$$
