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.

The jpegs did not come out well.... in Modern Elementary Theory of Numbers by Leonard Eugene Dickson, (1939) , on the top of page 161, homework exercise 2 is this: given that $C$ is a positive integer, $$ x^2 + 2 y^2 - C z^2 $$ is universal if and only if $C$ is odd and every prime factor of $C$  is $\; \equiv 1 \; \mbox{or} \; 3 \pmod 8$ 
Dickson and his students Ross and Oppenheim found all universal indefinite ternary quadratic forms, collected into four types (up to $SL_3 \mathbb Z$ equivalence of forms). Take $M$ any integer and $N$ any odd integer, not necessarily positive in either case. In all four cases it it obvious that they are universal, just experiment a bit with each. In the first, take $(x,1,0)$ for example. 
$$ xy - M z^2 $$
$$ 2xy - N z^2 $$
$$ 2xy + y^2- N z^2 $$
$$ 2xy + y^2- 2 N z^2 $$
In the fourth form, (I) for odd numbers $(x,1,0);$ (II) for numbers $2 \pmod 4$ take $(x,2,1);$ (III) for numbers $0 \pmod 4$ take $(x,2,0).$
Notice there is no $x^2$ term in any of these, so that the form evaluated at $(1,0,0)$ comes out to $0. \;$ A fundamental part of this is that a universal (ternary) form must non-trivially represent $0.$ Not true for quaternaries, such as $w^2 - 2 x^2 - 3 y^2 + 6 z^2$
Meanwhile, your $x^2 + 2 y^2 - 3 z^2$ is equivalent to the fourth type listed above, namely $2xy+y^2 + 6 z^2.$ For quadratic forms, we take (half) the Hessian matrix of second partials; forms with such Gram matrices $G,H$ are equivalent when there is an integer matrix $P$ with $\det P = 1$ and $P^T GP = H.$ Notice that this means $Q = P^{-1}$ is also of integers with $\det Q = 1,$ while $Q^T H Q = G.$ I have a program that finds me such matrices $P.$ Equivalent forms integrally represent exactly the same values. 
The first matrix identity below reads $$ (x+y)^2 + 2 (x+3z)^2 - 3(x+2z)^2 = 2xy+y^2 + 6 z^2 $$
$$  $$
$$
\left(
\begin{array}{ccc}
1&1&1 \\
1&0&0 \\
0&3&2 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
1&0&0 \\
0&2&0 \\
0&0&-3 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
1&1&0 \\
1&0&3 \\
1&0&2 \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
0&1&0 \\
1&1&0 \\
0&0&6 \\
\end{array}
\right)
$$
$$
\left(
\begin{array}{ccc}
0&1&0 \\
-2&2&1 \\
3&-3&-1 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
0&1&0 \\
1&1&0 \\
0&0&6 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
0&-2&3 \\
1&2&-3 \\
0&1&-1 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
1&0&0 \\
0&2&0 \\
0&0&-3 \\
\end{array}
\right)
$$
For the second matrix identity, take $u = -2y+3z, \; v = x + 2 y - 3 z, \; w = y - z, \;$  giving
$$ 2uv+v^2 + 6 w^2 = x^2 + 2 y^2 - 3 z^2  $$
