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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. 24
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, 119, 142, 166, 191, 226, 262, 300, 354, 405, 467, 540, 623, 705, 807, 927, 1060, 1206, 1369, 1551, 1760, 1998, 2248, 2556, 2861, 3236, 3628, 4100, 4587, 5152, 5756 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.
LINKS
EXAMPLE
The a(3) = 1 through a(15) = 11 partitions (A=10, B=11, C=12):
21 . . 42 421 431 63 532 542 84 643 653 A5
321 521 432 541 632 642 742 743 843
621 631 821 651 841 752 942
721 5321 921 A21 761 C21
4321 5421 5431 842 6432
6321 6421 B21 6531
7321 5432 7431
6431 7521
6521 8421
7421 9321
8321 54321
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Select[Tuples[#, 3], #[[1]]+#[[2]]==#[[3]]&]!={}&]], {n, 0, 30}]
PROG
(Python)
from itertools import combinations_with_replacement
from collections import Counter
from sympy.utilities.iterables import partitions
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
CROSSREFS
For subsets of {1..n} we have A093971 (sum-full sets), complement A007865.
The non-strict version is A363225, ranks A364348 (complement A364347).
The complement is counted by A364346, non-strict A364345.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions not re-using parts, complement A237113.
A323092 counts double-free partitions, ranks A320340.
Sequence in context: A076492 A127462 A106436 * A075758 A125596 A351962
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 19 2023
EXTENSIONS
a(31)-a(56) from Chai Wah Wu, Sep 20 2023
STATUS
approved

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Last modified May 28 01:12 EDT 2024. Contains 372900 sequences. (Running on oeis4.)