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 A355393 #6 Aug 26 2022 16:52:36
%S A355393 1,0,1,2,3,4,6,7,10,14,17,23,32,39,51,67,83,105,134,165,206,256,312,
%T A355393 385,475,573,697,849,1021,1231,1483,1771,2121,2534,3007,3575,4245,
%U A355393 5008,5914,6979,8198,9626,11292,13201,15430,18010,20960,24389,28346,32855,38066
%N A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.
%C A355393 These are partitions without a neighborless singleton, where a part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.
%e A355393 The a(0) = 1 through a(8) = 10 partitions:
%e A355393 () . (11) (21) (22) (32) (33) (43) (44)
%e A355393 (111) (211) (221) (222) (322) (332)
%e A355393 (1111) (2111) (321) (2221) (2222)
%e A355393 (11111) (2211) (3211) (3221)
%e A355393 (21111) (22111) (3311)
%e A355393 (111111) (211111) (22211)
%e A355393 (1111111) (32111)
%e A355393 (221111)
%e A355393 (2111111)
%e A355393 (11111111)
%t A355393 Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]
%Y A355393 This is the singleton case of A355394, complement A356236.
%Y A355393 The complement is counted by A356235.
%Y A355393 These partitions are ranked by the complement of A356237.
%Y A355393 The strict case is A356606, complement A356607.
%Y A355393 A000041 counts integer partitions, strict A000009.
%Y A355393 A000837 counts relatively prime partitions, ranked by A289509.
%Y A355393 A007690 counts partitions with no singletons, complement A183558.
%Y A355393 Cf. A073491, A325160, A328171, A328172, A328187, A328220, A356233.
%K A355393 nonn
%O A355393 0,4
%A A355393 _Gus Wiseman_, Aug 26 2022