Number of ordered triples of distinct positive integers summing to n.

0, 0, 0, 0, 0, 0, 6, 6, 12, 18, 24, 30, 42, 48, 60, 72, 84, 96, 114, 126, 144, 162, 180, 198, 222, 240, 264, 288, 312, 336, 366, 390, 420, 450, 480, 510, 546, 576, 612, 648, 684, 720, 762, 798, 840, 882, 924, 966, 1014, 1056, 1104, 1152, 1200, 1248, 1302, 1350
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Last updated: Oct 20 2020.

For this entry, let:
c(n) = Number of non-unimodal triples of distinct positive integers summing to n.
d(n) = Number of unimodal triples of distinct positive integers summing to n.

a(n) = 6*A001399(n-6) = 6*A069905(n-3) = 6*A211540(n-1).
a(n) = c(n) + d(n).
4*A001399(n-6) counts the neither increasing nor decreasing case.
2*A001399(n-6) counts the either increasing or decreasing case.

The a(6) = 6 through a(8) = 12 triples:
  (1,2,3)  (1,2,4)  (1,2,5)  
  (1,3,2)  (1,4,2)  (1,3,4)  
  (2,1,3)  (2,1,4)  (1,4,3)  
  (2,3,1)  (2,4,1)  (1,5,2)  
  (3,1,2)  (4,1,2)  (2,1,5)  
  (3,2,1)  (4,2,1)  (2,5,1)  
                    (3,1,4)  
                    (3,4,1)  
                    (4,1,3)  
                    (4,3,1)  
                    (5,1,2)  
                    (5,2,1)

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n,{3}],UnsameQ@@#&]],{n,0,30}]

c(n) is the non-unimodal case.
d(n) is the unimodal case.
A000217(n-2) is the not necessarily strict version, ranked by A014311.
A001399(n-6) = A069905(n-3) = A211540(n-1) are the unordered version.
A032020 counts these compositions of any length, ranked by A233564.
A072574 and A216652 have this sequence as column k = 3.
A337453 ranks these compositions.
A001399(n-3) = A069905(n) = A211540(n+2) count 3-part partitions, ranked by A014612.
Cf. A007304, A014612, A072707, A128012, A156040, A332834, A333149.