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

%I A367217 #16 Feb 25 2025 14:34:45
%S A367217 0,0,1,3,6,12,24,46,87,164,308,577,1080,2021,3779,7058,13166,24533,
%T A367217 45674,84978,158026,293737,545747,1013467,1881032,3489303,6468910,
%U A367217 11985988,22195905,41080751,75994642,140514019,259693004,479749492,885910870,1635281386
%N A367217 Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.
%F A367217 a(n) = 2^n - A367216(n). - _Chai Wah Wu_, Nov 14 2023
%e A367217 The a(2) = 1 through a(5) = 12 subsets:
%e A367217   {2}  {2}    {2}    {2}
%e A367217        {3}    {3}    {3}
%e A367217        {1,3}  {4}    {4}
%e A367217               {1,3}  {5}
%e A367217               {1,4}  {1,3}
%e A367217               {3,4}  {1,4}
%e A367217                      {1,5}
%e A367217                      {3,4}
%e A367217                      {3,5}
%e A367217                      {4,5}
%e A367217                      {1,4,5}
%e A367217                      {2,4,5}
%t A367217 Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,15}]
%Y A367217 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.
%Y A367217                sum-full   sum-free   comb-full  comb-free
%Y A367217               -------------------------------------------
%Y A367217   partitions:  A367212    A367213    A367218    A367219
%Y A367217   strict:      A367214    A367215    A367220    A367221
%Y A367217   subsets:     A367216    A367217*   A367222    A367223
%Y A367217   ranks:       A367224    A367225    A367226    A367227
%Y A367217 A000009 counts subsets summing to n.
%Y A367217 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A367217 A229816 counts partitions whose length is not a part, complement A002865.
%Y A367217 A007865/A085489/A151897 count certain types of sum-free subsets.
%Y A367217 A088809/A093971/A364534 count certain types of sum-full subsets.
%Y A367217 A124506 appears to count combination-free subsets, differences of A326083.
%Y A367217 A237667 counts sum-free partitions, ranks A364531.
%Y A367217 Triangles:
%Y A367217 A046663 counts partitions of n without a subset-sum k, strict A365663.
%Y A367217 A365381 counts sets with a subset summing to k, without A366320.
%Y A367217 A365541 counts sets containing two distinct elements summing to k.
%Y A367217 Cf. A068911, A103580, A240861, A288728, A326080, A326083, A364346, A364349, A365046, A365376, A365377.
%K A367217 nonn
%O A367217 0,4
%A A367217 _Gus Wiseman_, Nov 12 2023
%E A367217 a(16)-a(28) from _Chai Wah Wu_, Nov 14 2023
%E A367217 a(29)-a(35) from _Max Alekseyev_, Feb 25 2025