This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first terms, then the 9000 first terms.

550 http://imageshack.us/photo/my-images/11/golayrudinshapiro550.jpg/ 9000 http://imageshack.us/photo/my-images/11/golayrudinshapiro9000.jpg/

It makes sense to define a certain fractal $F$ as the "limit" of the graph $\Gamma=\{(n,a_n)\}_{n\ge0}$.
More precisely:
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.
Then because of the geometrical (almost-) similarity of the $\Gamma^0_k$, the limit $F:=\lim\limits_{k\to\infty}\Gamma^0_k\subset\mathbb R^2$ is well-defined. Note that its Hausdorff dimension is $d=2$.

Other examples:

• the sequence oeis 4074 that defines likewise the Blancmange_curve, dimension $d=1$
• sequences linked to the Gray code, like 3188 or 6068, both with $d=1$
• Stern's diatomic series yields a fractal with dimension $d=\frac{\ln 3}{\ln 2}$
• it makes sense to relate (if not to identify) the Cantor set with the sequence $1,0,1,0,0,0,1,0,1,... $ where $a_n=1$ iff the ternary representation of $n$ has only 0's and 2's (equivalently, the cellular automaton where at each step $1 \mapsto 101$ and $0 \mapsto000$), and to say this sequence has dimension $\frac{\ln2}{\ln3}$. Likewise for the "fat Cantor set" iterating 11100111 (dimension $\frac{\ln5}{\ln8}$) and all other sorts of Cantor dust.
• the devil's staircase, obtained by "integrating" the Cantor set, corresponds to this sequence, and a "mirrored" version of it can be found here.
• Other sequences of Toothpick and Cellular Automata type

Note that it is not at all straightforward or even possible to define a fractal for every self-similar integer sequence $a=(a_n)_{n\ge0}$ (self-similar meaning as usual that there is a $k\ge2$ and $\lambda$ such that $a_n=\lambda a_{kn}$). On the other hand, there are also sequences with a fractal-like appearance without being self-similar in the above sense.

Question:

• Has the idea of the "fractal dimension" of certain sequences been treated before?