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
KEYWORD
nonn,easy
AUTHOR
Gus Wiseman, May 15 2019
STATUS
approved