A364346 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.

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

Offset

0,5

Links

Table of n, a(n) for n=0..58.

Example

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

Mathematica

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

Prog

(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

Crossrefs

For subsets of {1..n} we have A007865 (sum-free sets), differences A288728.

For sums of any length > 1 we have A364349, non-strict A237667.

The complement is counted by A363226, non-strict A363225.

The non-strict version is A364345, ranks A364347, complement A364348.

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.

Cf. A002865, A025065, A085489, A093971, A108917, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

Sequence in context: A336315 A145394 A179806 * A182762 A173997 A029143

Adjacent sequences: A364343 A364344 A364345 * A364347 A364348 A364349

Keyword

nonn

Author

Gus Wiseman, Jul 22 2023

Status

approved