%I #15 Oct 18 2023 04:43:41
%S 1,1,2,2,4,5,7,10,13,16,21,27,34,43,54,67,83,102,122,151,182,218,258,
%T 313,366,443,513,611,713,844,975,1149,1325,1554,1780,2079,2381,2761,
%U 3145,3647,4134,4767,5408,6200,7014,8035,9048,10320,11639,13207,14836,16850
%N 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.
%e The a(1) = 1 through a(8) = 13 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (111) (22) (32) (33) (43) (44)
%e (31) (41) (51) (52) (53)
%e (1111) (311) (222) (61) (62)
%e (11111) (411) (322) (71)
%e (3111) (331) (332)
%e (111111) (511) (611)
%e (4111) (2222)
%e (31111) (3311)
%e (1111111) (5111)
%e (41111)
%e (311111)
%e (11111111)
%t Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,30}]
%Y For subsets of {1..n} instead of partitions we have A007865 (sum-free sets), differences A288728.
%Y Without re-using parts we have A236912, complement A237113.
%Y Allowing the sum of any number of parts gives A237667 (cf. A108917).
%Y The complement is counted by A363225, strict A363226, for subsets A093971.
%Y The strict case is A364346.
%Y These partitions have ranks A364347, complement A364348.
%Y A000041 counts partitions, strict A000009.
%Y A008284 counts partitions by length, strict A008289.
%Y A323092 counts double-free partitions, ranks A320340.
%Y Cf. A002865, A025065, A026905, A111133, A240861, A275972, A320347, A325862, A326083, A363260.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jul 20 2023