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A356607 Number of strict integer partitions of n with at least one neighborless part.

%I A356607 #20 Feb 12 2024 16:32:02
%S A356607 0,1,1,1,2,2,3,4,6,6,9,11,13,17,20,24,30,36,41,52,60,71,84,100,114,
%T A356607 137,158,183,214,248,283,330,379,432,499,570,648,742,846,955,1092,
%U A356607 1234,1395,1580,1786,2005,2270,2548,2861,3216,3610,4032,4526,5055,5642,6304,7031,7820,8720,9694
%N A356607 Number of strict integer partitions of n with at least one neighborless part.
%C A356607 A part x is neighborless if neither x - 1 nor x + 1 are parts.
%H A356607 Alois P. Heinz, <a href="/A356607/b356607.txt">Table of n, a(n) for n = 0..5000</a> (first 101 terms from Lucas A. Brown)
%H A356607 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A356607.py">A356607.py</a>
%e A356607 The a(0) = 0 through a(9) = 6 partitions:
%e A356607   .  (1)  (2)  (3)  (4)   (5)   (6)   (7)    (8)    (9)
%e A356607                     (31)  (41)  (42)  (52)   (53)   (63)
%e A356607                                 (51)  (61)   (62)   (72)
%e A356607                                       (421)  (71)   (81)
%e A356607                                              (431)  (531)
%e A356607                                              (521)  (621)
%t A356607 Table[Length[Select[IntegerPartitions[n],Function[ptn,UnsameQ@@ptn&&Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]
%Y A356607 This is the strict case of A356235 and A356236.
%Y A356607 The complement is counted by A356606, non-strict A355393 and A355394.
%Y A356607 A000041 counts integer partitions, strict A000009.
%Y A356607 A000837 counts relatively prime partitions, ranked by A289509.
%Y A356607 A007690 counts partitions with no singletons, complement A183558.
%Y A356607 Cf. A073492, A137921, A325160, A328171, A328172, A328187, A328220, A328221.
%K A356607 nonn
%O A356607 0,5
%A A356607 _Gus Wiseman_, Aug 26 2022
%E A356607 a(31)-a(59) from _Lucas A. Brown_, Sep 09 2022