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A363225 Number of integer partitions of n containing three parts (a,b,c) (repeats allowed) such that a + b = c. A variation of sum-full partitions.

%I A363225 #18 Mar 23 2024 22:12:51
%S A363225 0,0,0,1,1,2,4,5,9,14,21,29,43,58,81,109,148,195,263,339,445,574,744,
%T A363225 942,1209,1515,1923,2399,3005,3721,4629,5693,7024,8589,10530,12804,
%U A363225 15596,18876,22870,27538,33204,39816,47766,57061,68161,81099,96510,114434,135634
%N A363225 Number of integer partitions of n containing three parts (a,b,c) (repeats allowed) such that a + b = c. A variation of sum-full partitions.
%C A363225 Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.
%e A363225 The a(3) = 1 through a(9) = 14 partitions:
%e A363225   (21)  (211)  (221)   (42)     (421)     (422)      (63)
%e A363225                (2111)  (321)    (2221)    (431)      (432)
%e A363225                        (2211)   (3211)    (521)      (621)
%e A363225                        (21111)  (22111)   (3221)     (3321)
%e A363225                                 (211111)  (4211)     (4221)
%e A363225                                           (22211)    (4311)
%e A363225                                           (32111)    (5211)
%e A363225                                           (221111)   (22221)
%e A363225                                           (2111111)  (32211)
%e A363225                                                      (42111)
%e A363225                                                      (222111)
%e A363225                                                      (321111)
%e A363225                                                      (2211111)
%e A363225                                                      (21111111)
%t A363225 Table[Length[Select[IntegerPartitions[n],Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]!={}&]],{n,0,15}]
%o A363225 (Python)
%o A363225 from collections import Counter
%o A363225 from itertools import combinations_with_replacement
%o A363225 from sympy.utilities.iterables import partitions
%o A363225 def A363225(n): return sum(1 for p in partitions(n) if any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # _Chai Wah Wu_, Sep 21 2023
%Y A363225 For subsets of {1..n} we have A093971, A088809 without re-using parts.
%Y A363225 The complement for subsets is A007865, A085489 without re-using parts.
%Y A363225 Without re-using parts we have A237113, complement A236912.
%Y A363225 For sums of any length > 1 (without re-usable parts) we have A237668, complement A237667.
%Y A363225 The strict case is A363226.
%Y A363225 The complement is counted by A364345, strict A364346.
%Y A363225 These partitions have ranks A364348, complement A364347.
%Y A363225 The strict linear combination-free version is A364350.
%Y A363225 A000041 counts partitions, strict A000009.
%Y A363225 A008284 counts partitions by length, strict A008289.
%Y A363225 A323092 counts double-free partitions, ranks A320340.
%Y A363225 Cf. A002865, A025065, A026905, A108917, A237984, A326083, A363260.
%K A363225 nonn
%O A363225 0,6
%A A363225 _Gus Wiseman_, Jul 19 2023
%E A363225 a(31)-a(48) from _Chai Wah Wu_, Sep 21 2023