A364345 Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.

1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 27, 34, 43, 54, 67, 83, 102, 122, 151, 182, 218, 258, 313, 366, 443, 513, 611, 713, 844, 975, 1149, 1325, 1554, 1780, 2079, 2381, 2761, 3145, 3647, 4134, 4767, 5408, 6200, 7014, 8035, 9048, 10320, 11639, 13207, 14836, 16850

Offset

0,3

Links

Table of n, a(n) for n=0..51.

Example

The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (611)
                                               (4111)     (2222)
                                               (31111)    (3311)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Select[Tuples[Union[#], 3], #[[1]]+#[[2]]==#[[3]]&]=={}&]], {n, 0, 30}]

Crossrefs

For subsets of {1..n} instead of partitions we have A007865 (sum-free sets), differences A288728.

Without re-using parts we have A236912, complement A237113.

Allowing the sum of any number of parts gives A237667 (cf. A108917).

The complement is counted by A363225, strict A363226, for subsets A093971.

The strict case is A364346.

These partitions have ranks A364347, complement A364348.

A000041 counts partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A323092 counts double-free partitions, ranks A320340.

Cf. A002865, A025065, A026905, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

Sequence in context: A211863 A333190 A098859 * A239455 A362610 A363260

Adjacent sequences: A364342 A364343 A364344 * A364346 A364347 A364348

Keyword

nonn

Author

Gus Wiseman, Jul 20 2023

Status

approved