A295341 The number of partitions of n in which at least one part is a multiple of 3.

0, 0, 0, 1, 1, 2, 4, 6, 9, 14, 20, 29, 41, 57, 78, 106, 142, 189, 250, 327, 425, 549, 705, 900, 1144, 1445, 1819, 2279, 2844, 3534, 4379, 5403, 6648, 8152, 9969, 12152, 14780, 17920, 21682, 26163, 31504, 37842, 45371, 54270, 64800, 77211, 91842, 109031, 129235, 152897

Offset

0,6

Comments

From Gus Wiseman, May 23 2022: (Start)

Also the number of integer partitions of n with at least one part appearing more than twice. The Heinz numbers of these partitions are given by A046099. For example, the a(0) = 0 though a(8) = 9 partitions are:

. . . (111) (1111) (2111) (222) (2221) (2222)

(11111) (3111) (4111) (5111)

(21111) (22111) (22211)

(111111) (31111) (32111)

(211111) (41111)

(1111111) (221111)

(311111)

(2111111)

(11111111)

(End)

Links

Table of n, a(n) for n=0..49.

Formula

a(n) = A000041(n)-A000726(n).

Example

From Gus Wiseman, May 23 2022: (Start)
The a(0) = 0 through a(8) = 9 partitions with a part that is a multiple of 3:
  .  .  .  (3)  (31)  (32)   (6)     (43)     (53)
                      (311)  (33)    (61)     (62)
                             (321)   (322)    (332)
                             (3111)  (331)    (431)
                                     (3211)   (611)
                                     (31111)  (3221)
                                              (3311)
                                              (32111)
                                              (311111)
(End)

Mathematica

Table[Length[Select[IntegerPartitions[n], MemberQ[#/3, _?IntegerQ]&]], {n, 0, 30}] (* Gus Wiseman, May 23 2022 *)
Table[Length[Select[IntegerPartitions[n], MatchQ[#, {___, x_, x_, x_, ___}]&]], {n, 0, 30}] (* Gus Wiseman, May 23 2022 *)

Crossrefs

The complement is counted by A000726, ranked by A004709.

These partitions are ranked by A354235.

This is column k = 3 of A354234.

For 2 instead of 3 we have A047967, ranked by A013929 and A324929.

For 4 instead of 3 we have A295342, ranked by A046101.

A000041 counts integer partitions, strict A000009.

A046099 lists non-cubefree numbers.

Cf. A001522, A006918, A064410, A064428, A117485, A188674, A325187, A325534.

Sequence in context: A034748 A069916 A153140 * A139135 A097197 A260600

Adjacent sequences: A295338 A295339 A295340 * A295342 A295343 A295344

Keyword

nonn

Author

R. J. Mathar, Nov 20 2017

Status

approved