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A364346
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Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.
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26
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1, 1, 1, 1, 2, 3, 2, 4, 4, 5, 5, 8, 9, 11, 11, 16, 16, 20, 20, 25, 30, 34, 38, 42, 50, 58, 64, 73, 80, 90, 105, 114, 128, 148, 158, 180, 201, 220, 241, 277, 306, 333, 366, 404, 447, 497, 544, 592, 662, 708, 797, 861, 954, 1020, 1131, 1226, 1352, 1456, 1600
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OFFSET
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0,5
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LINKS
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EXAMPLE
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The a(1) = 1 through a(14) = 11 partitions (A..E = 10..14):
1 2 3 4 5 6 7 8 9 A B C D E
31 32 51 43 53 54 64 65 75 76 86
41 52 62 72 73 74 93 85 95
61 71 81 82 83 A2 94 A4
531 91 92 B1 A3 B3
A1 543 B2 C2
641 732 C1 D1
731 741 652 851
831 751 932
832 941
931 A31
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&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 A364346(n): return sum(1 for p in partitions(n) if max(p.values(), default=1)==1 and not any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()), 3))) # Chai Wah Wu, Sep 20 2023
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CROSSREFS
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For subsets of {1..n} we have A007865 (sum-free sets), differences A288728.
A236912 counts sum-free partitions not re-using parts, complement A237113.
Cf. A002865, A025065, A085489, A093971, A108917, A111133, A240861, A275972, A320347, A325862, A326083, A363260.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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