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 A367394 #11 Nov 20 2023 08:14:13
%S A367394 0,0,1,0,1,1,3,3,6,7,14,15,25,30,46,54,80,97,139,169,229,282,382,461,
%T A367394 607,746,962,1173,1499,1817,2302,2787,3467,4201,5216,6260,7702,9261,
%U A367394 11294,13524,16418,19572,23658,28141,33756,40081,47949,56662,67493,79639
%N A367394 Number of integer partitions of n whose length is a semi-sum of the parts.
%C A367394 We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
%e A367394 For the partition y = (3,3,2,1) we have 4 = 3 + 1, so y is counted under a(9).
%e A367394 The a(2) = 1 through a(10) = 14 partitions:
%e A367394 (11) . (211) (221) (321) (421) (521) (621) (721)
%e A367394 (2211) (2221) (2222) (3222) (3322)
%e A367394 (3111) (3211) (3221) (3321) (3331)
%e A367394 (3311) (4221) (4222)
%e A367394 (32111) (4311) (4321)
%e A367394 (41111) (32211) (5221)
%e A367394 (42111) (5311)
%e A367394 (32221)
%e A367394 (33211)
%e A367394 (42211)
%e A367394 (43111)
%e A367394 (331111)
%e A367394 (421111)
%e A367394 (511111)
%t A367394 Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,10}]
%Y A367394 The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
%Y A367394 sum-full sum-free comb-full comb-free semi-full semi-free
%Y A367394 -----------------------------------------------------------
%Y A367394 partitions: A367212 A367213 A367218 A367219 A367394* A367398
%Y A367394 strict: A367214 A367215 A367220 A367221 A367395 A367399
%Y A367394 subsets: A367216 A367217 A367222 A367223 A367396 A367400
%Y A367394 ranks: A367224 A367225 A367226 A367227 A367397 A367401
%Y A367394 A000041 counts partitions, strict A000009.
%Y A367394 A002865 counts partitions whose length is a part, complement A229816.
%Y A367394 A236912 counts partitions containing no semi-sum, ranks A364461.
%Y A367394 A237113 counts partitions containing a semi-sum, ranks A364462.
%Y A367394 A237668 counts sum-full partitions, sum-free A237667.
%Y A367394 A366738 counts semi-sums of partitions, strict A366741.
%Y A367394 Triangles:
%Y A367394 A008284 counts partitions by length, strict A008289.
%Y A367394 A365543 counts partitions with a subset-sum k, strict A365661.
%Y A367394 A367404 counts partitions with a semi-sum k, strict A367405.
%Y A367394 Cf. A000700, A088809, A093971, A126796, A238628, A304792, A363225, A364534, A365541, A365924, A367402.
%K A367394 nonn
%O A367394 0,7
%A A367394 _Gus Wiseman_, Nov 19 2023