  Explicit dyonic dilaton black holes of the four-dimensional
Einstein-Maxwell-dilaton theory are known only for two particular values of the
dilaton coupling constant $a =1,\sqrt{3}$, while for other $a$ numerical
evidence was presented earlier about existence of extremal dyons in theories
with the discrete sequence of dilaton couplings $a=\sqrt{n(n+1)/2}$ with
integer $n$. Apart from the lower members $n=1,\,2$, this family of theories
does not have motivation from supersymmetry or higher dimensions, and so far
the above quantization rule has not been derived analytically. We fill this gap
showing that this rule follows from analyticity of the dilaton at the
$AdS_2\times S^2$ event horizon with $n$ being the leading dilaton power in the
series expansion. We also present generalization for asymptotically anti-de
Sitter dyonic black holes with spherical, plane and hyperbolic topology of the
horizon.
