It makes sense to define a certain fractal $F$ as the "limit" of the graph $\Gamma=\{(n,a_n)\}_{n\ge0}$.
Fix a rectangle $R\subset\mathbb R^2$, e.g. the unit square. Take the part $\Gamma_k$ of $\Gamma$ between $n=2^{2k-1}$ and $n=2^{2k+1}-1$ and rescale it to a graph $\Gamma^0_k$ that fits $R$ best.