A337484 Number of ordered triples of positive integers summing to n that are neither strictly increasing nor strictly decreasing.

0, 0, 0, 1, 3, 6, 8, 13, 17, 22, 28, 35, 41, 50, 58, 67, 77, 88, 98, 111, 123, 136, 150, 165, 179, 196, 212, 229, 247, 266, 284, 305, 325, 346, 368, 391, 413, 438, 462, 487, 513, 540, 566, 595, 623, 652, 682, 713, 743, 776, 808, 841, 875, 910, 944, 981, 1017

Offset: 0

Author

Gus Wiseman, Sep 11 2020

Keywords

• nonn

Examples

			The a(3) = 1 through a(7) = 13 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,3,2)  (1,3,3)
           (2,1,1)  (1,3,1)  (1,4,1)  (1,4,2)
                    (2,1,2)  (2,1,3)  (1,5,1)
                    (2,2,1)  (2,2,2)  (2,1,4)
                    (3,1,1)  (2,3,1)  (2,2,3)
                             (3,1,2)  (2,3,2)
                             (4,1,1)  (2,4,1)
                                      (3,1,3)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,1,2)
                                      (5,1,1)
		

Crossrefs

A140106 is the unordered case.

A242771 allows strictly increasing but not strictly decreasing triples.

A337481 counts these compositions of any length.

A001399(n - 6) counts unordered strict triples.

A001523 counts unimodal compositions, with complement A115981.

A007318 and A097805 count compositions by length.

A069905 counts unordered triples.

A218004 counts strictly increasing or weakly decreasing compositions.

A332745 counts partitions with weakly increasing or weakly decreasing run-lengths.

A332835 counts compositions with weakly increasing or weakly decreasing run-lengths.

A337483 counts triples either weakly increasing or weakly decreasing.

Cf. A000212, A000217, A001840, A014311, A046691, A128422, A156040, A332834, A337461, A337482, A337561, A337603, A337604.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&&!Greater@@#&]],{n,0,15}]

Formula

a(n) = 2*A242771(n - 1) - A000217(n - 1), n > 0.

2*A001399(n - 6) = 2*A069905(n - 3) = 2*A211540(n - 1) is the complement.

4*A001399(n - 6) = 4*A069905(n - 3) = 4*A211540(n - 1) is the strict case.

Conjectures from Colin Barker, Sep 13 2020: (Start)

G.f.: x^3*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6.

(End)