A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

0, 0, 0, 1, 0, 2, 2, 4, 3, 7, 6, 10, 9, 14, 13, 19, 17, 24, 23, 30, 28, 37, 35, 44, 42, 52, 50, 61, 58, 70, 68, 80, 77, 91, 88, 102, 99, 114, 111, 127, 123, 140, 137, 154, 150, 169, 165, 184, 180, 200, 196, 217, 212, 234, 230, 252, 247, 271, 266, 290, 285, 310

Offset

0,6

Comments

Confirmed recurrence relation from Colin Barker for n <= 10000. - Fausto A. C. Cariboni, Feb 19 2022

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

Formula

Conjectures from Colin Barker, May 15 2019: (Start)

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

Example

The a(3) = 1 through a(13) = 14 partitions (A = 10, B = 11):
  (111)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)   (544)
         (311)  (411)  (331)  (521)  (432)  (442)  (533)  (543)   (553)
                       (421)  (611)  (441)  (622)  (542)  (552)   (643)
                       (511)         (522)  (631)  (551)  (732)   (652)
                                     (531)  (721)  (632)  (741)   (661)
                                     (621)  (811)  (641)  (822)   (733)
                                     (711)         (722)  (831)   (742)
                                                   (731)  (921)   (751)
                                                   (821)  (A11)   (832)
                                                   (911)          (841)
                                                                  (922)
                                                                  (931)
                                                                  (A21)
                                                                  (B11)

Mathematica

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

Crossrefs

Column k = 3 of A325592.

Cf. A000041, A001399, A005044, A008642, A069905, A266223.

Cf. A325689, A325691, A325694.

Sequence in context: A096441 A100824 A163227 * A238779 A239832 A240010

Adjacent sequences: A325687 A325688 A325689 * A325691 A325692 A325693

Keyword

nonn

Author

Gus Wiseman, May 15 2019

Status

approved