A365827 Number of strict integer partitions of n whose length is not 2.

1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 25, 30, 38, 45, 55, 66, 79, 93, 111, 130, 153, 179, 209, 242, 282, 325, 375, 432, 496, 568, 651, 742, 846, 963, 1094, 1240, 1406, 1589, 1795, 2026, 2282, 2567, 2887, 3240, 3634, 4072, 4557, 5094, 5692, 6351

Offset: 0

Author

Gus Wiseman, Sep 20 2023

Keywords

• nonn

Comments

Also the number of strict integer partitions of n with no pair of distinct parts summing to n.

Examples

			The a(5) = 1 through a(13) = 12 strict partitions (A..D = 10..13):
  (5)  (6)    (7)    (8)    (9)    (A)     (B)     (C)     (D)
       (321)  (421)  (431)  (432)  (532)   (542)   (543)   (643)
                     (521)  (531)  (541)   (632)   (642)   (652)
                            (621)  (631)   (641)   (651)   (742)
                                   (721)   (731)   (732)   (751)
                                   (4321)  (821)   (741)   (832)
                                           (5321)  (831)   (841)
                                                   (921)   (931)
                                                   (5421)  (A21)
                                                   (6321)  (5431)
                                                           (6421)
                                                           (7321)
		

Crossrefs

The complement is counted by A140106 shifted left.

Heinz numbers are A005117 \ A006881 = A005117 /\ A100959.

The non-strict version is A058984, complement A004526.

The case not containing n/2 is A365826, non-strict A365825.

A000041 counts integer partitions, strict A000009.

A046663 counts partitions with no submultiset summing to k, strict A365663.

A182616 counts partitions of 2n that do not contain n, strict A365828.

Cf. A008967, A035363, A078408, A365659.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[#]!=2&]],{n,0,30}]

Formula

a(n) = A000009(n) - A004526(n-1) for n > 0.