cp's OEIS Frontend

A363226 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.

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