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A367217
Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.
24
0, 0, 1, 3, 6, 12, 24, 46, 87, 164, 308, 577, 1080, 2021, 3779, 7058, 13166, 24533, 45674, 84978, 158026, 293737, 545747, 1013467, 1881032, 3489303, 6468910, 11985988, 22195905
OFFSET
0,4
FORMULA
a(n) = 2^n - A367216(n). - Chai Wah Wu, Nov 14 2023
EXAMPLE
The a(2) = 1 through a(5) = 12 subsets:
{2} {2} {2} {2}
{3} {3} {3}
{1,3} {4} {4}
{1,3} {5}
{1,4} {1,3}
{3,4} {1,4}
{1,5}
{3,4}
{3,5}
{4,5}
{1,4,5}
{2,4,5}
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#], Length[#]]&]], {n, 0, 15}]
CROSSREFS
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free
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A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A229816 counts partitions whose length is not a part, complement A002865.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A237667 counts sum-free partitions, ranks A364531.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts sets containing two distinct elements summing to k.
Sequence in context: A163314 A018183 A196787 * A200662 A371792 A264507
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 12 2023
EXTENSIONS
a(16)-a(28) from Chai Wah Wu, Nov 14 2023
STATUS
approved