This is inspired by the self-similarity of the celebrated [Golay-Rudin-Shapiro sequence][1], more exactly, of its alternating partial sums. (This latter one is [oeis 020990][2]). The pictures show the 550 first terms, then the 9000 first terms.

![550][3]
![9000][4]

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 $d=3/2$.

Other examples: 

  - the sequence [oeis 004074][5] that defines likewise the [Blancmange curve][6], dimension $d=1$
  - sequences linked to the Gray code, like [003188][7] or [006068][8], both with $d=1$ 
  - [Stern's diatomic series][9] (a.k.a. Stern-Brocot sequence or [$fusc$ function][10]) yields a fractal with dimension $d=\frac{\ln 3}{\ln 2}$
  - it makes sense to relate (if not to *identify*) the [Cantor set][11] 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][12], obtained by "integrating" the Cantor set, corresponds to [this sequence][13], and a "mirrored" version of it can be found [here.][14]  

  - Other sequences of [Toothpick and Cellular Automata type][15]

Like for most other sequences of this kind, the ressemblance is best seen when looking at a range from either $1$ to $2^n$, or (for some, like the Blancmange curve) from $2^{n-1}$ to $2^n$.  

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 investigated before?***


  [1]: https://en.wikipedia.org/wiki/Rudin-Shapiro_sequence
  [2]: http://oeis.org/A020990
  [3]: https://i.sstatic.net/ZPZTD.jpg
  [4]: https://i.sstatic.net/WDU1V.jpg
  [5]: http://oeis.org/A004074/graph
  [6]: http://en.wikipedia.org/wiki/Blancmange_curve
  [7]: http://oeis.org/A003188/graph
  [8]: http://oeis.org/A006068/graph
  [9]: http://oeis.org/A002487/graph
  [10]: https://en.wikipedia.org/wiki/Calkin-Wilf_tree#Stern's_diatomic_sequence
  [11]: https://en.wikipedia.org/wiki/Cantor_set
  [12]: http://en.wikipedia.org/wiki/Cantor_function
  [13]: http://oeis.org/A081611/graph
  [14]: http://oeis.org/A005836/graph
  [15]: http://oeis.org/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS