Triangular distribution of a+b sums for rationals a/b of bounded height Let $a/b$ be a rational in $(0,1)$ expressed in lowest terms.
The height of $a/b$ is $\max \{a,b\}$.
For each rational in $(0,1)$ of height $\le h$, form the sum $a+b$.
So, for example, here are the rationals of height $\le 6$
and their sums:
\begin{array}{ccccccccccc}
 \frac{1}{6} & \frac{1}{5} &
   \frac{1}{4} & \frac{1}{3} &
   \frac{2}{5} & \frac{1}{2} &
   \frac{3}{5} & \frac{2}{3} &
   \frac{3}{4} & \frac{4}{5} &
   \frac{5}{6} \\
 7 & 6 & 5 & 4 & 7 & 3 & 8 & 5
   & 7 & 9 & 11 \\
\end{array}
Now form a histogram of these sums. So above, the bin for $3$ gets $1$ count
because only $\frac{1}{2}$ leads to $3$, but the bin for $7$ gets a count
of $3$ because each of $\{ \frac{1}{6}, \frac{2}{5}, \frac{3}{4} \}$ sum to $7$.
Here is the histogram for $h \le 24$:

          


You can see the bin for $3$ still has a count of $1$, the
bin for $47$ has a count of $1$ for $\frac{23}{24}$, and the bin
for $46$ is empty. The tallest bin is for $23$, whose sum is
achieved by $11$ fractions: $\{ \frac{1}{22}, \frac{2}{21}, \ldots, \frac{11}{12} \}$.

Here is the histogram for $h \le 256$:




My question is: What explains the features of this plot.

I would appreciate explanations of some of the structure that seems to be
emerging in this plot,
from the left-right symmetry to more subtle features.
Perhaps I am hallucinating structure where there
is none, but it seems one can discern a series of nested triangles that roughly
demarcate regions of differing density, something like this:

          



Effectively, you have a histogram of $a + b$ for all $a < b \leq n$ such that $\gcd(a, b) = 1$.
For $a = 1$, we get all $b > 1$.
For $a = 2$, we get roughly half of all  $b > 2$.
What does that look like when we add them together?

We continue like this, for $a = 3$, roughly one third, etc.
For $a \leq 5$, we get this:

Does this illuminate the pattern?
