  Rueppel's conjecture on the linear complexity of the first $n$ terms of the
sequence $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots)$ was first proved by Dai using
the Euclidean algorithm. We have previously shown that we can attach a
homogeneous (annihilator) ideal of $F[x,z]$ to the first $n$ terms of a
sequence over a field $F$ and construct a pair of generating forms for it. This
approach gives another proof of Rueppel's conjecture. We also prove additional
properties of these forms and deduce the outputs of the LFSR synthesis
algorithm applied to the first $n$ terms. Further, dehomogenising the leading
generators yields the minimal polynomials of Dai.
