A337483 Number of ordered triples of positive integers summing to n that are either weakly increasing or weakly decreasing.

0, 0, 0, 1, 2, 4, 5, 8, 10, 13, 16, 20, 23, 28, 32, 37, 42, 48, 53, 60, 66, 73, 80, 88, 95, 104, 112, 121, 130, 140, 149, 160, 170, 181, 192, 204, 215, 228, 240, 253, 266, 280, 293, 308, 322, 337, 352, 368, 383, 400, 416, 433, 450, 468, 485, 504, 522, 541, 560

Offset: 0

Author

Gus Wiseman, Sep 07 2020

Keywords

• nonn
• easy

Examples

			The a(3) = 1 through a(8) = 10 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (2,1,1)  (1,2,2)  (1,2,3)  (1,2,4)  (1,2,5)
                    (2,2,1)  (2,2,2)  (1,3,3)  (1,3,4)
                    (3,1,1)  (3,2,1)  (2,2,3)  (2,2,4)
                             (4,1,1)  (3,2,2)  (2,3,3)
                                      (3,3,1)  (3,3,2)
                                      (4,2,1)  (4,2,2)
                                      (5,1,1)  (4,3,1)
                                               (5,2,1)
                                               (6,1,1)
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

A001399(n - 3) = A069905(n) = A211540(n + 2) counts the unordered case.

2*A001399(n - 6) = 2*A069905(n - 3) = 2*A211540(n - 1) counts the strict case.

A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts the strict unordered case.

A329398 counts these compositions of any length.

A218004 counts strictly increasing or weakly decreasing compositions.

A337484 counts neither strictly increasing nor strictly decreasing compositions.

Cf. A000212, A000217, A001840, A014311, A156040, A337461, A337603, A337604.

Programs

Mathematica

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

Formula

a(n > 0) = 2*A001399(n - 3) - A079978(n).

From Colin Barker, Sep 08 2020: (Start)

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

E.g.f.: (36 - 9*exp(-x) + exp(x)*(6*x^2 + 6*x - 19) - 8*exp(-x/2)*cos(sqrt(3)*x/2))/36. - Stefano Spezia, Apr 05 2023