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

Author

Gus Wiseman, Jul 20 2023

Keywords

• nonn

Examples

			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)
		

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.

Programs

Mathematica

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