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 A237113 #17 Sep 17 2023 18:43:49
%S A237113 0,0,0,0,1,1,3,3,8,10,17,22,37,47,71,91,133,170,236,301,408,515,686,
%T A237113 860,1119,1401,1798,2232,2829,3495,4378,5381,6682,8165,10060,12238,
%U A237113 14958,18116,22018,26533,32071,38490,46265,55318,66193,78843,93949,111503,132326
%N A237113 Number of partitions of n such that some part is a sum of two other parts.
%C A237113 These are partitions containing the sum of some 2-element submultiset of the parts, a variation of binary sum-full partitions where parts cannot be re-used, ranked by A364462. The complement is counted by A236912. The non-binary version is A237668. For re-usable parts we have A363225. - _Gus Wiseman_, Aug 10 2023
%F A237113 a(n) = A000041(n) - A236912(n).
%e A237113 Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1]. Thus, a(6) = 3.
%e A237113 From _Gus Wiseman_, Aug 09 2023: (Start)
%e A237113 The a(0) = 0 through a(9) = 10 partitions:
%e A237113 . . . . (211) (2111) (321) (3211) (422) (3321)
%e A237113 (2211) (22111) (431) (4221)
%e A237113 (21111) (211111) (3221) (4311)
%e A237113 (4211) (5211)
%e A237113 (22211) (32211)
%e A237113 (32111) (42111)
%e A237113 (221111) (222111)
%e A237113 (2111111) (321111)
%e A237113 (2211111)
%e A237113 (21111111)
%e A237113 (End)
%t A237113 z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]], Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
%t A237113 u = PartitionsP[Range[z]] - t (* A237113, _Peter J. C. Moses_, Feb 03 2014 *)
%t A237113 Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,30}] (* _Gus Wiseman_, Aug 09 2023 *)
%Y A237113 The complement for subsets is A085489, with re-usable parts A007865.
%Y A237113 For subsets of {1..n} we have A088809, with re-usable parts A093971.
%Y A237113 The complement is counted by A236912, ranks A364461.
%Y A237113 The non-binary complement is A237667, ranks A364531.
%Y A237113 The non-binary version is A237668, ranks A364532.
%Y A237113 With re-usable parts we have A363225, ranks A364348.
%Y A237113 The complement with re-usable parts is A364345, ranks A364347.
%Y A237113 These partitions have ranks A364462.
%Y A237113 The strict case is A364670, with re-usable parts A363226.
%Y A237113 A000041 counts integer partitions, strict A000009.
%Y A237113 A008284 counts partitions by length, strict A008289.
%Y A237113 A108917 counts knapsack partitions, ranks A299702.
%Y A237113 A323092 counts double-free partitions, ranks A320340.
%Y A237113 Cf. A002865, A151897, A237984, A325862, A326083, A363260.
%K A237113 nonn
%O A237113 0,7
%A A237113 _Clark Kimberling_, Feb 04 2014
%E A237113 a(0)=0 prepended by _Alois P. Heinz_, Sep 17 2023