%I #18 Oct 18 2023 04:44:04
%S 0,0,0,1,0,0,2,1,2,3,5,4,6,7,11,11,16,18,26,29,34,42,51,62,72,84,101,
%T 119,142,166,191,226,262,300,354,405,467,540,623,705,807,927,1060,
%U 1206,1369,1551,1760,1998,2248,2556,2861,3236,3628,4100,4587,5152,5756
%N Number of strict integer partitions of n containing some three possibly equal parts (a,b,c) such that a + b = c. A variation of sum-full strict partitions.
%C Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.
%e The a(3) = 1 through a(15) = 11 partitions (A=10, B=11, C=12):
%e 21 . . 42 421 431 63 532 542 84 643 653 A5
%e 321 521 432 541 632 642 742 743 843
%e 621 631 821 651 841 752 942
%e 721 5321 921 A21 761 C21
%e 4321 5421 5431 842 6432
%e 6321 6421 B21 6531
%e 7321 5432 7431
%e 6431 7521
%e 6521 8421
%e 7421 9321
%e 8321 54321
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]!={}&]],{n,0,30}]
%o (Python)
%o from itertools import combinations_with_replacement
%o from collections import Counter
%o from sympy.utilities.iterables import partitions
%o def A363226(n): return sum(1 for p in partitions(n) if max(p.values(),default=0)==1 and any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # _Chai Wah Wu_, Sep 20 2023
%Y For subsets of {1..n} we have A093971 (sum-full sets), complement A007865.
%Y The non-strict version is A363225, ranks A364348 (complement A364347).
%Y The complement is counted by A364346, non-strict A364345.
%Y A000041 counts partitions, strict A000009.
%Y A008284 counts partitions by length, strict A008289.
%Y A236912 counts sum-free partitions not re-using parts, complement A237113.
%Y A323092 counts double-free partitions, ranks A320340.
%Y Cf. A002865, A025065, A026905, A085489, A108917, A237667, A237668 A240861, A275972, A320347, A326083.
%K nonn
%O 0,7
%A _Gus Wiseman_, Jul 19 2023
%E a(31)-a(56) from _Chai Wah Wu_, Sep 20 2023