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%I #18 Sep 17 2020 20:34:39
%S 0,0,0,1,3,6,8,13,17,22,28,35,41,50,58,67,77,88,98,111,123,136,150,
%T 165,179,196,212,229,247,266,284,305,325,346,368,391,413,438,462,487,
%U 513,540,566,595,623,652,682,713,743,776,808,841,875,910,944,981,1017
%N Number of ordered triples of positive integers summing to n that are neither strictly increasing nor strictly decreasing.
%F a(n) = 2*A242771(n - 1) - A000217(n - 1), n > 0.
%F 2*A001399(n - 6) = 2*A069905(n - 3) = 2*A211540(n - 1) is the complement.
%F 4*A001399(n - 6) = 4*A069905(n - 3) = 4*A211540(n - 1) is the strict case.
%F Conjectures from _Colin Barker_, Sep 13 2020: (Start)
%F G.f.: x^3*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
%F a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6.
%F (End)
%e The a(3) = 1 through a(7) = 13 triples:
%e (1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5)
%e (1,2,1) (1,2,2) (1,3,2) (1,3,3)
%e (2,1,1) (1,3,1) (1,4,1) (1,4,2)
%e (2,1,2) (2,1,3) (1,5,1)
%e (2,2,1) (2,2,2) (2,1,4)
%e (3,1,1) (2,3,1) (2,2,3)
%e (3,1,2) (2,3,2)
%e (4,1,1) (2,4,1)
%e (3,1,3)
%e (3,2,2)
%e (3,3,1)
%e (4,1,2)
%e (5,1,1)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&&!Greater@@#&]],{n,0,15}]
%Y A140106 is the unordered case.
%Y A242771 allows strictly increasing but not strictly decreasing triples.
%Y A337481 counts these compositions of any length.
%Y A001399(n - 6) counts unordered strict triples.
%Y A001523 counts unimodal compositions, with complement A115981.
%Y A007318 and A097805 count compositions by length.
%Y A069905 counts unordered triples.
%Y A218004 counts strictly increasing or weakly decreasing compositions.
%Y A332745 counts partitions with weakly increasing or weakly decreasing run-lengths.
%Y A332835 counts compositions with weakly increasing or weakly decreasing run-lengths.
%Y A337483 counts triples either weakly increasing or weakly decreasing.
%Y Cf. A000212, A000217, A001840, A014311, A046691, A128422, A156040, A332834, A337461, A337482, A337561, A337603, A337604.
%K nonn
%O 0,5
%A _Gus Wiseman_, Sep 11 2020
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