This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A295341 #16 May 24 2022 08:13:27
%S A295341 0,0,0,1,1,2,4,6,9,14,20,29,41,57,78,106,142,189,250,327,425,549,705,
%T A295341 900,1144,1445,1819,2279,2844,3534,4379,5403,6648,8152,9969,12152,
%U A295341 14780,17920,21682,26163,31504,37842,45371,54270,64800,77211,91842,109031,129235,152897
%N A295341 The number of partitions of n in which at least one part is a multiple of 3.
%C A295341 From _Gus Wiseman_, May 23 2022: (Start)
%C A295341 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:
%C A295341 . . . (111) (1111) (2111) (222) (2221) (2222)
%C A295341 (11111) (3111) (4111) (5111)
%C A295341 (21111) (22111) (22211)
%C A295341 (111111) (31111) (32111)
%C A295341 (211111) (41111)
%C A295341 (1111111) (221111)
%C A295341 (311111)
%C A295341 (2111111)
%C A295341 (11111111)
%C A295341 (End)
%F A295341 a(n) = A000041(n)-A000726(n).
%e A295341 From _Gus Wiseman_, May 23 2022: (Start)
%e A295341 The a(0) = 0 through a(8) = 9 partitions with a part that is a multiple of 3:
%e A295341 . . . (3) (31) (32) (6) (43) (53)
%e A295341 (311) (33) (61) (62)
%e A295341 (321) (322) (332)
%e A295341 (3111) (331) (431)
%e A295341 (3211) (611)
%e A295341 (31111) (3221)
%e A295341 (3311)
%e A295341 (32111)
%e A295341 (311111)
%e A295341 (End)
%t A295341 Table[Length[Select[IntegerPartitions[n],MemberQ[#/3,_?IntegerQ]&]],{n,0,30}] (* _Gus Wiseman_, May 23 2022 *)
%t A295341 Table[Length[Select[IntegerPartitions[n],MatchQ[#,{___,x_,x_,x_,___}]&]],{n,0,30}] (* _Gus Wiseman_, May 23 2022 *)
%Y A295341 The complement is counted by A000726, ranked by A004709.
%Y A295341 These partitions are ranked by A354235.
%Y A295341 This is column k = 3 of A354234.
%Y A295341 For 2 instead of 3 we have A047967, ranked by A013929 and A324929.
%Y A295341 For 4 instead of 3 we have A295342, ranked by A046101.
%Y A295341 A000041 counts integer partitions, strict A000009.
%Y A295341 A046099 lists non-cubefree numbers.
%Y A295341 Cf. A001522, A006918, A064410, A064428, A117485, A188674, A325187, A325534.
%K A295341 nonn
%O A295341 0,6
%A A295341 _R. J. Mathar_, Nov 20 2017