  Given a prime $p$, an elliptic curve $\E/\F_p$ over the finite field $\F_p$
of $p$ elements and a binary \lrs\ $\(u(n)\)_{n =1}^\infty$ of order~$r$, we
study the distribution of the sequence of points $$ \sum_{j=0}^{r-1} u(n+j)P_j,
\qquad n =1,..., N, $$ on average over all possible choices of $\F_p$-rational
points $P_1,..., P_r$ on~$\E$. For a sufficiently large $N$ we improve and
generalise a previous result in this direction due to E.~El~Mahassni.
