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

0,5

Links

Table of n, a(n) for n=0..56.

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)

Example

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)

Mathematica

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

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.

Sequence in context: A046670 A131383 A219730 * A139001 A090961 A073355

Adjacent sequences: A337481 A337482 A337483 * A337485 A337486 A337487

Keyword

nonn

Author

Gus Wiseman, Sep 11 2020

Status

approved