Which integers can be written as $x^2+2y^2-3z^2\ $? 
For which integers $n$ has the diophantine equation $$x^2+2y^2-3z^2=n$$ solutions ?

These theorems
https://en.wikipedia.org/wiki/15_and_290_theorems
do not apply because the given quadratic form is not positive (or negative) definite. It seems that the quadratic form is universal (for every integer $n$ a solution exists) , but I have no idea how this can be proven.

We claim that any $n\in\mathbb{Z}$ can be written as $x^2+2y^2-3z^2$. 
Note that any such integer $n$ is $0$ or it is  equal to a $4^a\cdot 2^b\cdot d$ where $a$ is a non negative integer, $b\in\{0,1\}$ and $d$ is a signed odd number. Then we consider the following cases.
0) If $n=0$ then let $x=0$, $y=0$ and $z=0$.
1) If $n=d=2k+1$ then let $x=k+1$, $y=k$ and $z=k$:
$$x^2+2y^2-3z^2=(k+1)^2-k^2=2k+1=n.$$
2) If $n=2d=2(2k+1)$ then let $x=k$, $y=k+1$ and $z=k$:
$$x^2+2y^2-3z^2=2(k+1)^2-2k^2=4k+2=n.$$
3) If $n=4^ak$ and $k$ can be written as $x_k^2+2y_k^2-3z_k^2$ then
let $x=2^ax_k$, $y=2^ay_k$ and $z=2^az_k$:
$$x^2+2y^2-3z^2=4^a(x_k^2+2y_k^2-3z_k^2)=4^ak=n.$$
