cp's OEIS Frontend

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.

%I A364345 #15 Oct 18 2023 04:43:41
%S A364345 1,1,2,2,4,5,7,10,13,16,21,27,34,43,54,67,83,102,122,151,182,218,258,
%T A364345 313,366,443,513,611,713,844,975,1149,1325,1554,1780,2079,2381,2761,
%U A364345 3145,3647,4134,4767,5408,6200,7014,8035,9048,10320,11639,13207,14836,16850
%N 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.
%e A364345 The a(1) = 1 through a(8) = 13 partitions:
%e A364345   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A364345        (11)  (111)  (22)    (32)     (33)      (43)       (44)
%e A364345                     (31)    (41)     (51)      (52)       (53)
%e A364345                     (1111)  (311)    (222)     (61)       (62)
%e A364345                             (11111)  (411)     (322)      (71)
%e A364345                                      (3111)    (331)      (332)
%e A364345                                      (111111)  (511)      (611)
%e A364345                                                (4111)     (2222)
%e A364345                                                (31111)    (3311)
%e A364345                                                (1111111)  (5111)
%e A364345                                                           (41111)
%e A364345                                                           (311111)
%e A364345                                                           (11111111)
%t A364345 Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,30}]
%Y A364345 For subsets of {1..n} instead of partitions we have A007865 (sum-free sets), differences A288728.
%Y A364345 Without re-using parts we have A236912, complement A237113.
%Y A364345 Allowing the sum of any number of parts gives A237667 (cf. A108917).
%Y A364345 The complement is counted by A363225, strict A363226, for subsets A093971.
%Y A364345 The strict case is A364346.
%Y A364345 These partitions have ranks A364347, complement A364348.
%Y A364345 A000041 counts partitions, strict A000009.
%Y A364345 A008284 counts partitions by length, strict A008289.
%Y A364345 A323092 counts double-free partitions, ranks A320340.
%Y A364345 Cf. A002865, A025065, A026905, A111133, A240861, A275972, A320347, A325862, A326083, A363260.
%K A364345 nonn
%O A364345 0,3
%A A364345 _Gus Wiseman_, Jul 20 2023