A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 8, 11, 10, 13, 12, 15, 14, 18, 16, 20, 19, 23, 21, 26, 24, 29, 27, 32, 30, 36, 33, 39, 37, 43, 40, 47, 44, 51, 48, 55, 52, 60, 56, 64, 61, 69, 65, 74, 70, 79, 75, 84, 80, 90, 85, 95, 91, 101, 96, 107, 102, 113

Offset

0,7

Comments

Also the number of possible triples of edge-lengths of a triangle with perimeter n, where degenerate (self-intersecting) triangles are allowed.

The number of triples (a,b,c) for 1 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020

Links

Stefano Spezia, Table of n, a(n) for n = 0..10000

Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough, Torsors and tilings from toric toggling, arXiv:2305.07627 [math.CO], 2023. See g.f. at p. 20.

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

Formula

Conjectures from Colin Barker, May 16 2019: (Start)

G.f.: x^3*(1 + x - x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8. (End)

a(n) = A005044(n+3) - A000035(n+3). i.e., remove the only one triple (a=0,b,b) if n is even from the A005044 which is the number of triples (a,b,c) for 0 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020

The above conjectured formulas are true. - Stefano Spezia, May 19 2023

Example

The a(3) = 1 through a(12) = 6 partitions:
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)
                       (321)  (331)  (422)  (432)  (442)  (533)  (543)
                                     (431)  (441)  (532)  (542)  (552)
                                                   (541)  (551)  (633)
                                                                 (642)
                                                                 (651)

Mathematica

Table[Length[Select[IntegerPartitions[n, {3}], #[[1]]<=#[[2]]+#[[3]]&]], {n, 0, 30}]

Crossrefs

Cf. A001399, A005044 (nondegenerate triangles), A008642, A069905, A124278.

Cf. A325688, A325690, A325694.

Sequence in context: A363959 A366387 A083802 * A326668 A198318 A100881

Adjacent sequences: A325688 A325689 A325690 * A325692 A325693 A325694

Keyword

nonn,easy

Author

Gus Wiseman, May 15 2019

Status

approved