For quadratics with period-one negative continued fraction expansions,
X\theta =\frac{1}{ a-{\dfrac{\mathstrut 1}{a-{\dfrac{\mathstrut
1}{a- \cdots }}}}},
¶ we show that the inhomogeneous Lagrange spectrum,
\bL (\theta) :=\bigl\{ \liminf\nolimits_{|n|\rightarrow \infty}
|n|@\|n\theta -\gamma\| : \gamma \in \funnyR,\; \gamma \not\in
\funnyZ+\theta \funnyZ\bigr\},
¶ contains an inhomogeneous Hall's ray $[0,c(\theta)]$ with
$$c(\theta)=\tfrac{1}{4}\bigl(1-O(a^{-1/2})\bigr)\hbox{.}
¶ We describe gaps in the spectrum
showing that this is essentially best possible.
Pictures of computed spectra are included. Investigating such pictures
led us to these results.