a(6) = 13 because the sum of divisors of the 6th nontriangular (i.e., 9) is 1 + 3 + 9 = 13.
On the other we can see that in the main diagonal of the diagrams both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys as shown below.
Illustration of initial terms:
.
n m(n) a(n) Diagram
. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
_| | | | | | | | | | | | | | | | | | | | | | | | | | |
1 2 3 |_ _|_| | | | | | | | | | | | | | | | | | | | | | | | |
_ _| _|_| | | | | | | | | | | | | | | | | | | | | | |
2 4 7 |_ _ _| _|_| | | | | | | | | | | | | | | | | | | | |
3 5 6 |_ _ _| _| _ _|_| | | | | | | | | | | | | | | | | | |
_ _ _ _| _| | _ _|_| | | | | | | | | | | | | | | | |
4 7 8 |_ _ _ _| |_ _|_| _ _|_| | | | | | | | | | | | | | |
5 8 15 |_ _ _ _ _| _| | _ _ _|_| | | | | | | | | | | | |
6 9 13 |_ _ _ _ _| | _|_| | _ _ _|_| | | | | | | | | | |
_ _ _ _ _ _| _ _| _| | _ _ _|_| | | | | | | | |
7 11 12 |_ _ _ _ _ _| | _| _| _| | _ _ _ _|_| | | | | | |
8 12 28 |_ _ _ _ _ _ _| |_ _| _| _ _| | | _ _ _ _|_| | | | |
9 13 14 |_ _ _ _ _ _ _| | _ _| _| _| | | _ _ _ _|_| | |
10 14 24 |_ _ _ _ _ _ _ _| | | | _|_| | _ _ _ _ _|_|
_ _ _ _ _ _ _ _| | _ _| _ _|_| | | |
11 16 31 |_ _ _ _ _ _ _ _ _| | _ _| _| _ _|_| |
12 17 18 |_ _ _ _ _ _ _ _ _| | |_ _ _| _| | _ _|
13 18 39 |_ _ _ _ _ _ _ _ _ _| | _ _| _| _|_|
14 19 20 |_ _ _ _ _ _ _ _ _ _| | | |_ _|
15 20 42 |_ _ _ _ _ _ _ _ _ _ _| | _ _ _| _|
_ _ _ _ _ _ _ _ _ _ _| | | _ _| |
16 22 36 |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _|
17 23 24 |_ _ _ _ _ _ _ _ _ _ _ _| | |
18 24 60 |_ _ _ _ _ _ _ _ _ _ _ _ _| |
19 25 31 |_ _ _ _ _ _ _ _ _ _ _ _ _| |
20 26 42 |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
21 27 40 |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column m gives the nontriangular numbers.
Also the diagrams have on the main diagonal the following property: diagram [1] has peaks, diagrams [2, 3] have valleys, diagrams [4, 5, 6] have peaks, diagrams [7, 8, 9, 10] have valleys, and so on.
a(n) is also the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 3 + 3 = 6, so a(3) = 6.
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