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.

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 21, 29, 43, 58, 81, 109, 148, 195, 263, 339, 445, 574, 744, 942, 1209, 1515, 1923, 2399, 3005, 3721, 4629, 5693, 7024, 8589, 10530, 12804, 15596, 18876, 22870, 27538, 33204, 39816, 47766, 57061, 68161, 81099, 96510, 114434, 135634

Offset: 0

Author

Gus Wiseman, Jul 19 2023

Keywords

• nonn

Comments

Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.

Examples

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (431)      (432)
                       (2211)   (3211)    (521)      (621)
                       (21111)  (22111)   (3221)     (3321)
                                (211111)  (4211)     (4221)
                                          (22211)    (4311)
                                          (32111)    (5211)
                                          (221111)   (22221)
                                          (2111111)  (32211)
                                                     (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)
		

Crossrefs

For subsets of {1..n} we have A093971, A088809 without re-using parts.

The complement for subsets is A007865, A085489 without re-using parts.

Without re-using parts we have A237113, complement A236912.

For sums of any length > 1 (without re-usable parts) we have A237668, complement A237667.

The strict case is A363226.

The complement is counted by A364345, strict A364346.

These partitions have ranks A364348, complement A364347.

The strict linear combination-free version is A364350.

A000041 counts partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A323092 counts double-free partitions, ranks A320340.

Cf. A002865, A025065, A026905, A108917, A237984, A326083, A363260.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]!={}&]],{n,0,15}]

Python

from collections import Counter
from itertools import combinations_with_replacement
from sympy.utilities.iterables import partitions
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

Extensions

a(31)-a(48) from Chai Wah Wu, Sep 21 2023