

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 sumfull 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
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OFFSET

0,7


COMMENTS

Note that, by this definition, the partition (2,1) is sumfull, 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 (sumfull sets), complement A007865.
A236912 counts sumfree partitions not reusing parts, complement A237113.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



