What integers can be represented by the quadratic form $4x^2 - 3y^2 - z^2$?

$(x,y,z)=(2,1,1)$ is a solution of $4x^2-3y^2-z^2=12$, so it is solvable. This is probably quicker than factoring or reviewing some theory -- although there is a rich theory about ternary quadratic forms, and the question which integers they represent. Ramanujan studied this for the quadratic form $x^2+y^2+10z^2$. It is associated to the elliptic curve $y^2 = x^3+x^2+4x+4$, and the question which odd numbers are represented by it is very difficult, and most results here depend on generalised Riemann hypotheses. A nice article can be found here: http://www.stanford.edu/~rjlo/papers/09-TernaryQuadratics.pdf.