TITLE: Difference Triangles
QUESTION [9 upvotes]: In a difference triangle, a row of $n$ integers is given, then their differences are written underneath, and then another row of difference is added, until there is a triangle of $n (n+1)/2$ integers. For example:

The above triangle also has the property that it has 15 distinct values with no values from 1 to 15 either missing or repeated.  According to this query, this is the last difference triangle with nothing missing and nothing repeated.
That's similar to the last perfect ruler, with marks ${0,1,3,6}$. These marks measure all differences 1 to 6 with no lengths missing or repeated.  For longer rulers, there are two options.

No repeats allowed. The smallest ruler with $n$ marks is called a Golomb ruler.
No missing values. The longest ruler with $n$ marks is an optimal sparse ruler.

We can look at difference triangles with these same conditions.
Order 6.  $T_6 = 21$.
The smallest difference triangle with no repeats has large value 22 ($T_6  +1$) and misses 15.
The largest difference triangle with no misses has large value 20 ($T_6  -1$) and repeats 4.

Order 7.  $T_7=28$.
The smallest known difference triangle with no repeats has large value 33 ($T_7  +5$) and misses ${16, 20, 22, 29, 30}$.
The largest known difference triangle with no misses has large value 24 ($T_7  -4$) and repeats 1 and 3.

What happens with orders 8 and beyond?  With $T_8=36$, how close can we get to $36$ with either no repeats or no misses?

REPLY [2 votes]: (Not an answer. Simply short algorithm description and some of results)
If denote upper row of the triangle as
$$
(t_1, t_2, \ldots, t_{n-1}, t_n), \tag{1}
$$
then (w.l.o.g.) we'll consider only rows with $t_1 \le t_n$ (if $t_1=t_n$, then $t_2 \le t_{n-1}$ and so on).

To find largest difference triangle with no misses for relatively small $n$ $(n \le 9)$, one can apply exhaustive search for upper row (I'm unable to find smarter deterministic algorithm).

Rough estimation:

$n=7$:

up to $28^7 \approx 13.5 \times 10^9$ tests to check the value $28$ for optimality;

up to $27^7 \approx 10.5 \times 10^9$ tests to check the value $27$ for optimality;

...
$n=8$:

up to $36^8 \approx 2.8 \times 10^{12}$ tests to check the value $36$ for optimality;

up to $35^8 \approx 2.3 \times 10^{12}$ tests to check the value $35$ for optimality;

...
This way,

"$8$-no-misses" problem has three $30$-solutions (only upper rows are listed here):
(9, 28, 30, 5, 27, 24, 3, 29)

(16, 30, 28, 1, 5, 3, 29, 20)

(18, 4, 28, 26, 1, 30, 21, 27)

and there are no $k$-solutions for $k\ge 31$.
"$9$-no-misses" problem has at least one $38$-solution
(19, 35, 34, 3, 36, 38, 11, 37, 32)

(I didn't check it completely for this time).

To find smallest difference triangle with no repeats for relatively small $n$ $(n\le 10)$, I used recursive triangle construction.
Let $(n,k)$-no-repeats triangle is triangle with upper row $(1)$, where
$$t_j \le k, \; j=1,2,\ldots,n.$$
Then if $n$-triangle is $(n,k)$-no-repeats triangle, then its left $(n-1)$-triangle (with upper row $(t_1,t_2,\ldots,t_{n-1})$) has the same property:
 it is $(n-1,k)$-no-repeats triangle.
Which admits to apply recursion.
This way,

"$8$-no-repeats" problem has only one $44$-solution:
 (7, 33, 42, 3, 44, 43, 6, 29)

and no $k$-solutions for $k\le 43$.
"$9$-no-repeats" problem has only one $59$-solution:
 (17, 49, 58, 1, 55, 59, 13, 53, 43)

and no $k$-solutions for $k\le 58$.
"$10$-no-repeats" problem has only one $76$-solution:
 (45, 69, 12, 76, 73, 15, 75, 66, 16, 52) 

and no $k$-solutions for $k\le 75$.
"$11$-no-repeats" problem has at least one $102$-solution:
 (12, 77, 98, 19, 100, 95, 8, 102, 99, 16, 70) 

and no $k$-solutions for $k\le 78$ (currently checking cases $k\in[79 .. 102]$).

For larger $n$ heuristic or randomized algorithms can probably give (almost) optimal solutions.
Actually algorithm developed by Dmitry Kamenetsky works pretty nice and fast:
when I've seen his $(8,44)$ and $(9,59)$ (optimal!) results  for smallest difference triangle with no repeats, my "records" were $(8, 45)$ and $(9, 66)$ at that time.