cp's OEIS Frontend

A367216 Number of subsets of {1..n} whose cardinality is equal to the sum of some subset.

%I A367216 #18 Feb 25 2025 13:13:23
%S A367216 1,2,3,5,10,20,40,82,169,348,716,1471,3016,6171,12605,25710,52370,
%T A367216 106539,216470,439310,890550,1803415,3648557,7375141,14896184,
%U A367216 30065129,60639954,122231740,246239551,495790161,997747182,2006969629,4035274292,8110185100,16293958314,32724456982
%N A367216 Number of subsets of {1..n} whose cardinality is equal to the sum of some subset.
%F A367216 a(n) = 2^n - A367217(n). - _Chai Wah Wu_, Nov 14 2023
%e A367216 The a(0) = 1 through a(4) = 10 subsets:
%e A367216   {}  {}   {}     {}       {}
%e A367216       {1}  {1}    {1}      {1}
%e A367216            {1,2}  {1,2}    {1,2}
%e A367216                   {2,3}    {2,3}
%e A367216                   {1,2,3}  {2,4}
%e A367216                            {1,2,3}
%e A367216                            {1,2,4}
%e A367216                            {1,3,4}
%e A367216                            {2,3,4}
%e A367216                            {1,2,3,4}
%t A367216 Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]
%Y A367216 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 A367216                sum-full   sum-free   comb-full  comb-free
%Y A367216               -------------------------------------------
%Y A367216   partitions:  A367212    A367213    A367218    A367219
%Y A367216   strict:      A367214    A367215    A367220    A367221
%Y A367216   subsets:     A367216*   A367217    A367222    A367223
%Y A367216   ranks:       A367224    A367225    A367226    A367227
%Y A367216 A000009 counts subsets summing to n.
%Y A367216 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A367216 A002865 counts partitions whose length is a part, complement A229816.
%Y A367216 A007865/A085489/A151897 count certain types of sum-free subsets.
%Y A367216 A088809/A093971/A364534 count certain types of sum-full subsets.
%Y A367216 A237668 counts sum-full partitions, ranks A364532.
%Y A367216 A240855 counts strict partitions whose length is a part, complement A240861.
%Y A367216 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A367216 A365046 counts combination-full subsets, differences of A364914.
%Y A367216 Triangles:
%Y A367216 A365381 counts sets with a subset summing to k, without A366320.
%Y A367216 A365541 counts sets containing two distinct elements summing to k.
%Y A367216 Cf. A068911, A095944, A103580, A288728, A326080, A326083, A365376, A365544.
%K A367216 nonn
%O A367216 0,2
%A A367216 _Gus Wiseman_, Nov 12 2023
%E A367216 a(16)-a(28) from _Chai Wah Wu_, Nov 14 2023
%E A367216 a(29)-a(35) from _Max Alekseyev_, Feb 25 2025