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 A365825 #12 Sep 23 2023 12:11:43
%S A365825 1,1,1,2,2,5,6,12,14,26,31,51,61,95,114,169,202,289,347,481,576,782,
%T A365825 936,1244,1487,1946,2323,2997,3570,4551,5414,6827,8103,10127,11997,
%U A365825 14866,17575,21619,25507,31166,36692,44563,52362,63240,74152,89112,104281,124731
%N A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.
%C A365825 Also the number of integer partitions of n with no two possibly equal parts summing to n.
%F A365825 Heinz numbers are A100959 /\ !A344415.
%F A365825 a(n) = A000041(n)-(n-1)/2 if n is odd. a(n) = A000041(n)-n/2-A000041(n/2)+1 if n is even. - _Chai Wah Wu_, Sep 23 2023
%e A365825 The a(1) = 1 through a(8) = 14 partitions:
%e A365825 (1) (2) (3) (4) (5) (6) (7) (8)
%e A365825 (111) (1111) (221) (222) (322) (332)
%e A365825 (311) (411) (331) (521)
%e A365825 (2111) (2211) (421) (611)
%e A365825 (11111) (21111) (511) (2222)
%e A365825 (111111) (2221) (3221)
%e A365825 (3211) (3311)
%e A365825 (4111) (5111)
%e A365825 (22111) (22211)
%e A365825 (31111) (32111)
%e A365825 (211111) (221111)
%e A365825 (1111111) (311111)
%e A365825 (2111111)
%e A365825 (11111111)
%t A365825 Table[Length[Select[IntegerPartitions[n],Length[#]!=2&&FreeQ[#,n/2]&]],{n,0,15}]
%o A365825 (Python)
%o A365825 from sympy import npartitions
%o A365825 def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # _Chai Wah Wu_, Sep 23 2023
%Y A365825 First condition alone is A058984, complement A004526, ranks A100959.
%Y A365825 Second condition alone is A086543, complement A035363, ranks !A344415.
%Y A365825 The complement is counted by A238628.
%Y A365825 The strict case is A365826, complement A365659.
%Y A365825 A000041 counts integer partitions, strict A000009.
%Y A365825 A046663 counts partitions with no submultiset summing to k, strict A365663.
%Y A365825 A140106 counts strict partitions of length 2, complement A365827.
%Y A365825 A182616 counts partitions of 2n that do not contain n, strict A365828.
%Y A365825 Cf. A005408, A008967, A068911, A365377, A365544, A365543.
%K A365825 nonn
%O A365825 0,4
%A A365825 _Gus Wiseman_, Sep 19 2023
%E A365825 a(31)-a(47) from _Chai Wah Wu_, Sep 23 2023