cp's OEIS Frontend

A326080 Number of subsets of {1..n} containing the sum of every subset whose sum is <= n.

%I A326080 #13 Aug 30 2019 21:47:48
%S A326080 1,2,4,7,12,19,31,47,73,110,168,247,375,546,817,1193,1769,2552,3791,
%T A326080 5445,8012,11517,16899,24144,35391,50427,73614,104984,152656,216802,
%U A326080 315689,447473,648813,920163,1332991,1884735,2728020,3853437,5568644,7868096,11347437
%N A326080 Number of subsets of {1..n} containing the sum of every subset whose sum is <= n.
%C A326080 Equivalently, a(n) is the number of subsets of {1..n} containing the sum of any two distinct elements whose sum is <= n.
%C A326080 The summands must be distinct. The case where they are allowed to be equal is A326083.
%C A326080 If A151897 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).
%e A326080 The a(0) = 1 through a(5) = 19 subsets:
%e A326080   {}  {}   {}     {}       {}         {}
%e A326080       {1}  {1}    {1}      {1}        {1}
%e A326080            {2}    {2}      {2}        {2}
%e A326080            {1,2}  {3}      {3}        {3}
%e A326080                   {1,3}    {4}        {4}
%e A326080                   {2,3}    {1,4}      {5}
%e A326080                   {1,2,3}  {2,3}      {1,5}
%e A326080                            {2,4}      {2,4}
%e A326080                            {3,4}      {2,5}
%e A326080                            {1,3,4}    {3,4}
%e A326080                            {2,3,4}    {3,5}
%e A326080                            {1,2,3,4}  {4,5}
%e A326080                                       {1,4,5}
%e A326080                                       {2,3,5}
%e A326080                                       {2,4,5}
%e A326080                                       {3,4,5}
%e A326080                                       {1,3,4,5}
%e A326080                                       {2,3,4,5}
%e A326080                                       {1,2,3,4,5}
%e A326080 The a(6) = 31 subsets:
%e A326080   {}  {1}  {1,6}  {1,5,6}  {1,4,5,6}  {1,3,4,5,6}  {1,2,3,4,5,6}
%e A326080       {2}  {2,5}  {2,3,5}  {2,3,5,6}  {2,3,4,5,6}
%e A326080       {3}  {2,6}  {2,4,6}  {2,4,5,6}
%e A326080       {4}  {3,4}  {2,5,6}  {3,4,5,6}
%e A326080       {5}  {3,5}  {3,4,5}
%e A326080       {6}  {3,6}  {3,4,6}
%e A326080            {4,5}  {3,5,6}
%e A326080            {4,6}  {4,5,6}
%e A326080            {5,6}
%t A326080 Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[Plus@@@Subsets[#,{2}],#<=n&]]&]],{n,0,10}]
%o A326080 (PARI)
%o A326080 a(n)={
%o A326080    my(recurse(k, b)=
%o A326080       if( k > n, 1,
%o A326080           my(t=self()(k + 1, b + (1<<k)));
%o A326080           for(i=1, (k-1)\2, if(bittest(b, i) && bittest(b, k-i), return(t)));
%o A326080           t + self()(k + 1, b) )
%o A326080    );
%o A326080    recurse(1, 0);
%o A326080 } \\ _Andrew Howroyd_, Aug 30 2019
%Y A326080 Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326083.
%K A326080 nonn
%O A326080 0,2
%A A326080 _Gus Wiseman_, Jun 05 2019
%E A326080 Terms a(21) and beyond from _Andrew Howroyd_, Aug 30 2019