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A363225
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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.
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50
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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
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OFFSET
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0,6
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COMMENTS
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Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Select[Tuples[#, 3], #[[1]]+#[[2]]==#[[3]]&]!={}&]], {n, 0, 15}]
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PROG
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(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
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CROSSREFS
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For subsets of {1..n} we have A093971, A088809 without re-using parts.
The complement for subsets is A007865, A085489 without re-using parts.
For sums of any length > 1 (without re-usable parts) we have A237668, complement A237667.
The strict linear combination-free version is A364350.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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