A367217 Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.

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, 41080751, 75994642, 140514019, 259693004, 479749492, 885910870, 1635281386

Offset: 0

Author

Gus Wiseman, Nov 12 2023

Keywords

• nonn

Examples

			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}
		

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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partitions: A367212 A367213 A367218 A367219

strict: A367214 A367215 A367220 A367221

subsets: A367216 A367217* A367222 A367223

ranks: A367224 A367225 A367226 A367227

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.

Cf. A068911, A103580, A240861, A288728, A326080, A326083, A364346, A364349, A365046, A365376, A365377.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,15}]

Formula

a(n) = 2^n - A367216(n). - Chai Wah Wu, Nov 14 2023

Extensions

a(16)-a(28) from Chai Wah Wu, Nov 14 2023

a(29)-a(35) from Max Alekseyev, Feb 25 2025