It would be kind of cool to get a closed form for the number of integer solutions $$x^4+y^4-x^2y^2=n$$ which we will let $\phi_n$ denote.
It would be cool because we could exploit $\sum_{n=1}^N\phi_n\approx 2\sqrt{N}K(\frac{3}{4}) $ where $K$ is the complete elliptic integral of the first kind into giving us a representation of $$ \lim_{N\to\infty}\frac{\sqrt{N}}{N}\sum_{n=1}^N\phi_n=2K(\frac{3}{4}) $$
This is demonstrated here and here. The general technique here.
I am not sure we should expect a closed form for this $\phi_n$ but we can maybe hope that we can use something like theta series to make the job easier to look at this sequence. In the link in the previous sentence I demonstrate how we can use the product of these theta-like series to give us a generating function for the number of integer solutions to $x^4+y^4=n$ and a wider class of Diophantine curves. The technique doesn't seem to me to readily lend itself to getting a representation for the number of solutions to the Diophantine equation:
$$x^4-x^2y^2+y^4=n$$
It's clear that $\phi_0=1.$ It's easy to see that for $n\neq0$ we have $8|\phi_n$. So if $x=a,y=0$ satisfies the equation then so does $x=y=a$. Likewise if the equation is satisfied by $x=a, y=b$ where $a \neq b$ then clearly it is satisfied by $x=b, y=a$. It looks for most small $n$ we have $\phi_n \in 0,8$. I am curious if there is a value for which $\phi_n>8$
