About the equation $x^2+y^2+z^2+2t^2=n$ The question
The final goal (for this stage of my project) is to get an explicit form for $\phi(n)$. This last one is the number of integer solutions to $x^2+y^2+z^2+2t^2=n$. You may find this $\phi(n)$ on OEIS or you can find the first several numbers by pasting the following code into something like mathematica:
CoefficientList[(1 + 2 Sum[q^(2(j)^2), {j, 10}] ) *(1 + 2 Sum[q^((j)^2), {j, 10}] )^3 , q]


The question is what's an explicit form for $\phi(n)?$

Exposition
I would like to get an explicit formula for $\phi(n)$. Let me tell you what I know so far. It looks like $\phi(n)$ is well-behaved on odd $n$ and on powers of $2$.
$\chi(x)=  \sqrt{2}\sin\left(\frac{\pi}{4}x\right)\sin\left(\frac{\pi}{2}x\right)=\begin{cases} 1  \hspace{1 cm} \text{when }x \equiv 1,7 \mod 8 
\\ -1 \hspace{1 cm} \text{when }x \equiv 3,5 \mod 8  \end{cases}$
For $n\equiv 1, 7 \mod 8$ we have
$$\phi(n)=6\sum_{d|n} {\chi(d)}d$$
For example, $33$ is congruent to $1 \mod 8$ and indeed $\phi(33)=6\times(1-3-11+33)=120$
For $n \equiv 3,5 \mod 8$ we have
$$\phi(n)=-10\sum_{d|n} {\chi(d)}d$$
For powers of $2$ we have $\phi(2^\alpha)=2^{\alpha+3}-2$
I am using as a type of template here Joseph Liouville's Sur La Forme $x^2+y^2+z^2+3t^2$ and Sur La Forme $x^2+y^2+z^2+5t^2$. I am not able to complete the characterization and I am not $100 \%$ sure that Liouville does this for the forms above either(though I suspect he does) because I don't speak/read French very well (though math is math and this I can read). I can't find anything Liouville wrote on $x^2+y^2+z^2+2t^2$. If anyone knows that he surely did and can point me to the right spot I would appreciate it. Also if this is in Grosswald's text (which I don't own) I would definitely reconsider purchasing it. It's not clear to me from the TOC whether this is the right place to look. Also (and I sincerely doubt this one)... you know if Liouville is one of these mathematicians where everything has already been translated and I can find it English that would be amazing.
How do I finish the job and complete this characterization?
Motivations!
What I think would be very cool would be to argue that $ \sum_{n=1}^R \phi(n)$ is approximately the interior volume of $x^2+y^2+z^2+2t^2=R$ and thereby win a series of rational numbers that converges to the volume in the interior of $x^2+y^2+z^2+2t^2=1$ (which I would guess should be $\frac{\pi^2}{2\sqrt{2}}$? Correct me if I am wrong but this is kind of a minor detail.) I have explained that technique (perhaps ad nauseam)  over here.

Letting $\phi(n)$ denote the number of integer solutions to $x^2+y^2+z^2+2t^2$, with $n=2^{\alpha}N$ where $N$ is odd we have
$$\phi(n)=2\bigg( 2^{\alpha+2}\bigg( \frac{ 8}{N} \bigg)-1 \bigg) \sum_{d|N} \bigg( \frac{ 8}{d} \bigg)d$$
Where above appears Legendre-Jacobi-Kronecker symbols. In particular,
$$\bigg( \frac{8}{x} \bigg) = \cases{ \hspace{0.32 cm} 0  \text{ if } x \equiv 0 \hspace{.44 cm} (\mod 2) \\\hspace{0.32 cm} 1 \text{ if } x \equiv 1,7 (\mod 8) \\ -1 \text{ if } x \equiv 3,5 (\mod 8)}$$
An elementary proof can be found here.
So the next step in this process is to look for the asymptotic behavior of $\sum_{n=1}^x \phi(n)$ and see whether we can apply something like Abel's Summation here.
Indeed, we can conclude that $$\sum_{n=1}^\infty  \bigg(\frac{8}{n}\bigg) \frac{1}{n^2} =\frac{\pi^2}{8\sqrt2}$$
